Conference on Differential Geometry and Physics

Eotvos Lorand University, Budapest, Hungary29 August – 2 September, 2005

Abstracts of talksLast update: 11 August

Chern-Finsler connection and its applicationsTadashi Aikou (Department of Mathematics and Computer Science, Kagoshima

University)In this talk, we shall report the some properties of the Chern-Finsler connection∇ of a strongly pseudoconvex Finsler bundle (E,F ). In the case of E = TM ,the holomorphic tangent bundle of a complex manifold, we define the notion of(strongly) Finsler-Kahler manifold, and we shall show a characterization of Finsler-Kahler manifold.

Let G =∑

Gij(z, ζ)dzi ⊗ dzj be the complex Finsler metric on the pull-backbundle π∗TM , and ωTM =

√−1

∑Gijdzi ∧ dzj the Kahler form of G. Then we

say G is (strongly) Finsler-Kahler if it satisfies dHωTM ≡ 0, where dH means thederivation in the horizontal direction.

Since the metric G is the Hermitian metric on π∗TM , it defines a unique connec-tion, that is the Chern-Finsler connection ∇. If we denote by GR the real part ofG, ∇ induces a standard connection D on the real pull-back bundle π∗T RM whichsatisfies the metrical condition DGR ≡ 0 and the natural condition DJ ≡ 0 for thecomplex structure on π∗T RM .

On the other hand, the real part GR is an inner product on the bundle π∗T RM ,and so it defines a unique connection D on (π∗T RM,GR) which satisfies the metricalcondition DGR = 0 and some symmetric properties. We call this connection D theCartan connection. We note that D is not the Cartan connection in the usualsense, since the real part GR is not a real Finsler metric in the usual sense by Heil’sTherem. Then our main theorem is as follows:

Theorem. Let (M,F ) be a strongly pseudoconvex Finsler manifold. Then thefollowing conditions are equivalent.

(1) The Cartan connection D coincides with the Chern-Finsler connection ∇on π∗TM , i.e., D = D.

(2) The complex structure J is parallel with respect to D, i.e.,

DHJ = 0.

(3) The Kahler form ωTM is parallel with respect to D, i.e.,

DHωTM = 0.

(4) (M,F ) is a Finsler-Kahler manifold.

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The Holomorphic bisectional curvature of the complex Finsler spacesNicoleta Aldea (Transilvania University of Brasov)

The notion of holomorphic bisectional curvature for a complex Finsler space (M,F )is defined with respect to the Chern complex linear connection on the pull-back tan-gent bundle. By means of the holomorphic flag curvature of a complex Finsler space,a special approach is devoted to obtain the characterizations of the holomorphicbisectional curvature. For the class of generalized Einstein complex Finsler spacessome results concerning the holomorphic bisectional curvature are also given.

Surfaces in the Lorentz-Minkowski Space with Prescribed Gauss MapJuan A. Aledo (University of Castilla-La Mancha, Spain),

Jose M. Espinar and Jose A. Galvez (University of Granada, Spain)We obtain a Kenmotsu-type representation for timelike surfaces with prescribedGauss map in the 3-dimensional Lorentz-Minkowski space. We use such represen-tation in order to classify the complete timelike surfaces with positive constantGaussian curvature in this ambient space in terms of harmonic diffeomorphismsbetween simply connected Lorentz surfaces and the universal covering of the deSitter Space.

Complex product structures and associated foliations on 6-dimensionalnilpotent Lie groups

Adrian Andrada(The Abdus Salam International Centre for Theoretical Physics)

A complex product structure on a manifold is a pair J,E where J is a complexstructure and E is a product structure such that JE = −EJ . In this work westudy the class of invariant complex product structures on nilpotent Lie groups, i.e.the tensors J and E are invariant by left-translations of the group. From the generaltheory of complex product manifolds, we know that such a group admits foliationswhose leaves are totally geodesic and flat with respect to a canonical connectiondetermined by J and E; in the nilpotent case we can also prove that these leavesare complete and that this canonical connection is always Ricci-flat. In dimension6 we obtain the full classification of the Lie algebras of the Lie groups admitting acomplex product structure; all these groups admit discrete cocompact subgroups,and hence we obtain examples of complex product structures on 6-dimensionalnilmanifolds. We study the associated foliations in some particular Lie groups andnilmanifolds.

4-dimensional minimal CR submanifolds of the sphere S6

Miroslava Antic (Faculty of Mathematics, University of Belgrade),(Based on joint research with Mirjana Djoric and Luc Vrancken)

It is well known that sphere S6 admits an almost complex structure J constructedusing Cayley algebra, which is nearly Kaehler. Let M be a Riemannian submanifoldof the S6. M is a CR submanifold if there exists a C∞-differentiable holomorphicdistribution 4 ⊂ T (M) (i.e., J4 = 4) such that its orthogonal complement 4⊥

in T (M) is totally real (J4⊥ ⊆ N(M)), where N(M) is the normal bundle over

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M in S6. Since there is no 4-dimensional almost complex submanifold of S6 theholomorphic distribution has to be 2-dimensional.

We study 4-dimensional minimal CR submanifolds M of the nearly Kaehler 6-sphere S6(1) which satisfy Chen’s equality.

Classification of 4-dimensional homogeneous D’Atri spacesT. Arias-Marco (University of Valencia),O. Kowalski (Charles University, Prague)

The property of being a D’Atri space is equivalent to the infinite number of cur-vature identities called the odd Ledger conditions. In particular, a Riemannianmanifold (M, g) satisfying the first odd Ledger condition is said to be of type A.The first attempt to classify all homogeneous 4-dimensional D’Atri spaces was madein two papers by F. Podesta - A. Spiro and P. Bueken - L. Vanhecke (which aremutually complementary) but not in completely satisfactory way as we will show.Moreover, we present the complete classification of all homogeneous spaces of typeA in a simple and readable form and, as a result, we prove correctly that all homo-geneous 4-dimensional D’Atri spaces are naturally reductive.

Biharmonic Lorentz Hypersurfaces in E41

A. Arvanitoyeorgos (University of Patras Greece),F. Defever (Hogeschool Brugge-Oostende Belgium),G. Kaimakamis (Hellenic Army Academy Greece),B.J. Papantoniou (University of Patras Greece)

A submanifold Mnr of the pseudo-Euclidean space Es4 is said to have harmonic

mean curvature vector field if ∆ ~H = ~0, where ~H denotes the mean curvaturevector field, and ∆ the Laplacian of the induced pseudo-Riemannian metric. Weprove that every nondegenerate Lorentz hypersurface M3

1 of E41 with harmonic

mean curvature vector, is minimal.

Regularized volume forms of infinitedimensional manifolds and deRham type cohomology with infinite degree elements

Akira Asada (Former: Sinsyu University)Let H, g be a pair of a Hilbert space H and a positive Schatten class operator Gsuch that its ζ-function ζ(G, s) = tr(Gs) is holomorphic at s = 0. Then by usingζ(G, s), we can define ζ-regularization of infinite dimensional integrals on suitabledomain D in H, which gives a mathematical justification of appearence of the Ray-Singer determinant in Gaussian path integral. Precisely saying, regularized integralis performed on D, the extension of D to H, the total space of teh determinantbundle of H. Then we can calculate regualrized volume forms and volumes of infi-nite dimensiona sphere S∞ and torus T∞. By using regularized volume forms, wecan define de Rham type cohomology of S∞ and T∞, which allow Poincare duality.They have infinite degree elements. The de type cohomology of T∞ is isomorphicto the Grassmann algebra with infinite degree elements. Its multiplication rule de-pends on ν = ζ(G, 0). So it is not a topological invariant. Regularized volume formand de Rham type cohomology with infinite degree elements for general infinitedimensional manifolds are also treated.

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Weakly-Berwald spacesSandor Bacso (University of Debrecen)

Let (M,L) be a Finsler space with Berwald connection BΓ = (Ghk , Gh

jk), where Ghk

is the nonlinear connection naturally defined by the function L and Ghjk = ∂Gh

k/∂yj

are the connection coefficients of BΓ . The hv-curvature tensor of BΓ is defined byGh

ijk = ∂Ghjk/∂yi. The hv-Ricci curvature tensor Gij is defined by Gij =

∑r Gr

ijr.A Finsler space (M,L) is said to be Berwald if Gh

ijk ≡ 0, that is, if BΓ is a linearconnection, and (M,L) is said to be weakly-Berwald if Gij ≡ 0. Z. Shen introducedthe notion of weakly affine spray, and in accordance with this we gave the definitionof a weakly Berwald space. Wellknown example for weakly Berwald metric is theFunk metric. The main purpose of the present lecture is to investigate weaklyBerwald spaces. Our main result: a weakly Berwald space of scalar curvature is ofconstant curvature. Finally we will show some very simple conditions for Randersand Kropina spaces to be weakly Berwald spaces.

Crash-escape orbits of zero angular momentum in Schwarzschild metricand gravitational energy of a non-rotating perfect fluid shphere

Szabolcs Barcza (Konkoly Observatory of the Hungarian Academy of Sciences)The geodesic hypothesis is applied for infinitesimal spherical shells falling onto acentral mass, both having zero angular momentum. A scalar function is constructedto give the energy of the shell as a function of the distance from the central mass.An analysis of this process has shown that crossing of the Schwarzschild radius isallowed from geodetic point of view but yet it is forbidden from energetic point ofview. The gravitational energy of a non-rotating celestial body is derived.

Metric Rigidity of Holomorphic CurvesStefan Bechtluft-Sachs (University of Regensburg)

By a classical result of Calabi the first fundamental form determines a holomorphiccurve in complex projective space up to congruence. We extend this to curves fin more general Kahler manifolds, such as complex Grassmannians for instance. Inaddition to the first fundamental form, we need to prescribe certain invariants off , which are still intrinsically defined over M . Finally, we discuss extensions of thisprocedure to harmonic maps.

Metrics of open 5-dimensional open manifold of nonnegative curvatureand 3-dimensional soul

Mikael Bengtsson and Valery Marenich (University of Kalmar)In 2002, D. Gromoll and K. Tapp asked about a possible classification of nonnega-tively curved metrics on Sn×Rk, and pointed out that e.g. a nontrivial R2-bundleover S2 may have a very large family of such metrics (J. Cheeger and D. Gromoll,-72). The trivial bundle, however, is more restrictive. Some work have recentlybeen done, by K. Tapp and by V. Marenich, to describe trivial R3-bundles overS2. This case has been considered the most important 5-dimensional case, since itmight give some insights into the Hopf conjecture (whether S2×S2 admits a metricof positive curvature).

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My talk will be about the structure of the trivial R2 bundle over S3, and a bitabout possible metrics of nonnegative curvature, under the assumption that a cer-tain vector field has at least one zero, and is based on the article ”On nonnegativelycurved metrics on open five-dimensional manifolds” by Marenitch and Bengtsson.

Pairing in LF manifoldsMichel Nguifo Boyom (University Montpellier 2)

A locally flat manifold (LF manifold) (M,D) is a smooth manifold M endowedwith a torsion free linear connection D whose curvature tensor vanishes identically.Therefore the vector space X(M) of smooth vector fields on M is a Koszul-Vinbergalgebra. Let W be a left module of a Koszul-Vinberg algebra A and let W ′ be thedual vector space of W . There is a canonical pairing between the KV-homologyspace h(A,W ) and the KV-cohomology space H(A,W ′). I shall be concerned withthe case when A is the Koszul-Vinberg algebra X(M) of smooth vector fields on aLF manifold (M,D) and W is the vector space of smooth functions on the manifoldM .

Manifolds with Commuting Jacobi OperatorsMiguel Brozos-Vazquez (University of Santiago de Compostela, Spain),

Peter B. Gilkey ( University of Oregon, USA)The important role played by the Jacobi operator of a Riemannian manifold hasbeen emphasized during the last decade with the extensive work on the Ossermanproblem. Motivated by a work of Y. Tsankov where he study commutative Ja-cobi operators in hypersurfaces, we characterize Riemannian manifolds of constantsectional curvature in terms of commutation properties of their Jacobi operators,using similar technics to those developed in the study of the Osserman problem.The main result is as follows:

Theorem. Let (M, g) be a Riemannian manifold of dimension m ≥ 3.(1) If J(x)J(y) = J(y)J(x) for all x, y, then (M, g) is flat.(2) If J(x)J(y) = J(y)J(x) provided x ⊥ y, then (M, g) has constant sectional

curvature.

Geometry related with 3-forms of special type on 7-dimensionalmanifolds

Jarolım Bures (Mathematical Institute of Charles University, Prague)There are geometric structures related to some 3-forms on 7-dimensional manifoldswhich are of the same type at any point. In the seven-dimensional case we have eighttypes of (regular) forms corresponding of the orbits of Gl(7,R) on Λ3R7∗ representedalgebraically by some canonical ωi, 1 ≤ i ≤ 8 (see [BV1]). In the lecture geometricalstructures related to 3-forms of two special types which correspond to the two smallorbits inside Λ3R7∗ are studied. Its relation to other structures on 7-dimensionalmanifolds as 2-forms of maximal rank (presymplectic structures) and with specialattention to G2-structures, Sasakian structures, 3-Sasakian structures etc is given.There is strong relation with two papers by N.Hitchin (Hi1,Hi2). The structurediscussed here is also a special case of multisymplectic structures on manifoldsstudied by several authors (J.G. Marshden, H.J. Gotay, T.J. Bridges, M. de Leon,T. Catrijn and others) in relation to theoretical physics.

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References

[BV1] Bures J., Vanzura J., Multisymplectic forms of degree three in dimension seven, Proc. WS.Srni, 2002. Supp. Rend. Univ Salermo.

[BV2] Bures J., Vanzura J., Geometry of multisymplectic forms of degree three in dimensions six

and seven, preprint 2003.[CanIbLe] F. Cantrijn, A. Ibort, M. de Leon, On the geometry of multisymplectic manifolds,

preprint 2000.

[Hi1] N.Hitchin, The geometry of three-forms in six dimensions, J. Differential Geometry 55,(2000), 547-576.

[Hi2] N.Hitchin, Stable forms and special metrics, Contemporary Mathematics 288, 70-89, (2001).

H-contact unit tangent sphere bundlesG. Calvaruso and D. Perrone

(University of Lecce, Department of Mathematics ”E. De Giorgi”)

Recently, many authors investigated the harmonicity of unit vector fields in severalgeometric situations. In particular, an H-contact space is a contact metric mani-fold whose characteristic vector field ξ is harmonic. It is worthwhile to note thatthe class of H-contact metric manifolds extends several interesting classes of con-tact metric manifolds, like Sasakian, K-contact, (strongly) locally ϕ-symmetric and(k, µ)-spaces. We studied how the geometry of a Riemannian manifold (M, g) isinfluenced by the property that its unit tangent sphere bundle T1M , equipped withits standard contact metric structure, is H-contact. In several cases, we obtainedpositive answers to the basic question whether two-point homogeneous spaces arethe only Riemannian manifolds whose unit tangent sphere bundles are H-contactor not.

The isoperimetric problem in surfaces of revolutionAntonio Canete (Universidad de Granada)

Given a surface M , we may consider the isoperimetric problem consisting of findingthe least perimeter set in M enclosing a given area. Only for certain surfaces theisoperimetric solutions are completely known.

In this talk we will focus on several kinds of surfaces of revolution, as spheres ortori of revolution, where this problem has been recently solved.

Higher order jet involution and prolongation of connectionsMiroslav Doupovec (Brno University of Technology)

Wlodzimierz M. Mikulski (Jagiellonian University, Krakow)

We generalize the concept of a classical involution of the iterated tangent bundleto natural transformations of the composition of iterated functors. Then we solvethe following problems:

1. We introduce an exchange natural equivalence JrJs → JsJr between iteratedhigher order jet functors depending on a classical linear connection Λ on the basemanifold M .

2. We introduce an involution of non-holonomic jet spaces.3. We introduce the prolongation of higher order connections to jet bundles.

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References

[1] Ehresmann C., Sur les connexions d’ ordre superieur, Atti del V. Cong. del’ Unione Mat.Italiana, 1955, Roma Cremonese (1956), 344-346.

[2] Modugno M., Jet involution and prolongation of connections, Cas. Pest. Mat. 114 (1989),

356-365.

Deformation properties of one remarkable hypersurface in R4

Z. Dusek (Palacky University, Olomouc, Czech Republic)O. Kowalski (Charles University, Prague)

For the study of isometrical deformations of a hypersurface in Rn (n ≥ 4), theinteresting case is when the rank of the shape operator is equal to 2. In thiscase there are three possibilities: either the hypersurface is locally continuouslydeformable, or it is locally rigid, or, there is, up to a congruence, exactly onenontrivial local isometric deformation.We study the hypersurface M in R4 defined by the equation

f(x, y, x) =(x2 − y2)z − 2xy

2(z2 + 1).

This hypersurface was proposed by H. Takagi as a counterexample to the conjec-ture by K. Nomizu. The conjecture claims that every irreducible, complete semi-symmetric space of dimension greater than or equal to three is locally symmetric.We prove that this hypersurface is locally rigid. This solves an open problem stud-ied in the 70-ies by Japanese mathematicians.

Totally Umbilical Submanifolds in Pseudosymmetric and RelatedManifolds

Stanis law Ewert-Krzemieniewski(Institute of Mathematics, Szczecin University of Technology)

1. Basic Notations and Definitions

Suppose N be a manifold, n = dim N ≥ 3, g denotes a Riemannian or semi-Riemannian metric on N and ∇ be its Levi-Civita connection.

The Riemann curvature tensor R is an endomorphism of the Lie algebra of vectorfields on N defined by

R(X, Y )Z = ∇[X,Y ]Z − [∇X ,∇Y ]Z

and let R be its associated (0, 4) tensor

R(X, Y, Z, V ) = g(R(X, Y )Z, V )

For symmetric (0, 2) tensors A and B their Kulkarni-Nomizu product A ∧ B isgiven by

(A ∧B)(U,X, Y, V ) =

A(X, Y )B(U, V )−A(X, V )B(U, Y ) + A(U, V )B(X, Y )−A(U, Y )B(X, V ).

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Then the Weyl conformal curvature tensor C of type (0, 4) is defined as

C = R− 1n− 2

g ∧ S +r

2(n− 1)(n− 2)g ∧ g,

r being the scalar curvature of N.We extend the action of ∧ to the tensor B of type (0, 4) with symmetries

B(U,X, Y, V ) = B(X, U, Y, V ) = −B(U,X, V, Y )

setting

(A ∧B)(U,X, Y, V, Z, W ) =

A(X, Y )B(U, V, Z,W )−A(X, V )B(U, Y, Z, W )+

A(U, V )B(X, Y, Z,W )−A(U, Y )B(X, V, Z,W ).

For a skew-symmetric endomorphism P let P be a (0, 4) tensor associated withP by

P (X, Y, Z, V ) = g(P(X, Y )Z, V ).

One extends the endomorphism P(X, Y ) to a derivation P(X, Y )· of the Lie algebraof tensor fields on N assuming it commutes with contractions and setting:

P(X, Y ) · f = 0,

f being a function on N ;

(1) (P ·T )(X1, X2, . . . , Xk;X, Y ) = (P(X, Y ) · T )((X1, X2, . . . , Xk) =

− T (P(X, Y )X1, X2, . . . , Xk)− · · · − T (X1, . . . , Xk−1,P(X, Y )Xk),

T being a (0, k) tensor, k ≥ 1. In the case P =R we obtain the well known Ricciidentity.

In the same manner, for a symmetric tensor B of type (0, 2) and its associatedB,

B(X, Y ) = g(BX, Y )

we define B · T by

(B·T )(X1, X2, . . . , Xk) =

− T (BX1, X2, . . . , Xk)− · · · − T (X1, . . . , Xk−1,BXk).

For a (0, 2) tensor A on N we define

(X ∧A Y )Z = A(Y, Z)X −A(X, Z)Y.

If, moreover, A is symmetric and T is of type (0, k), k ≥ 1, we define the tensorsQ(A, T ) and Q(B,A, T ) of types (0, k + 2) setting

(2) Q(A, T )(X1, X2, . . . , Xk;X, Y ) = ((X ∧A Y ) · T )X1, X2, . . . , Xk) =

− T ((X ∧A Y )X1, X2, . . . , Xk)− · · · − T (X1, . . . Xk−1, (X ∧A Y )Xk)

and

Q(B,A, T )(X1, X2, . . . , Xk;X, Y ) = (B(X ∧A Y ) · T )(X1, X2, . . . , Xk).

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A manifold N is said to satisfy a curvature condition of pseudosymmetry type iftensor fields given by (1) and (2) are linearly dependent. On an appropriatelychosen generic subset of N this is equivalent to

P ·T = Q(A, T ).

2. Results

In the exposition we shell discus results concerning of totally umbilical subman-ifolds in manifolds satisfying one of the following conditions:

∇Y ∇XC − ∇X∇Y C = c(X, Y )C + A((X ∧g Y ) · C),

∇Y ∇XC − ∇X∇Y C = a(Y )∇XC − a(X)∇Y C + A((X ∧g Y ) · C),

for some function A on N , (0, 2) tensor field c and (0, 1) tensor field a,

R · C = L eCQ(g, C),

C · R = L eCQ(g, C),

C · C = L eCQ(g, C).Some examples will be given.

References

[B] M. Belkhelfa, R. Deszcz, M. G logowska, M. Hotlos, D. Kowalczyk, L. Verstraelen, On some

type of curvature conditions. In: PDEs, Submanifolds and Affine Differential Geometry,Banach Center Publication, vol. 57 (2002), 179-194.

[D1] R. Deszcz, On pseudosymmetric spaces, Bull. Soc.Math. Belg. Ser. A 44 (1992), 1-34.

[D2] R. Deszcz, Notes on totally umbilical submanifolds, Geometry and Topology of Submanifolds(Marseille, 1987), 89-97.

[DD] F. Defever, R. Deszcz, On Riemannian manifolds satisfying a certain curvature condition

imposed on the Weyl curvature tensor. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math.32 (1993), 27–34.

[V] L. Verstraelen, Comments on pseudo-symmetry in the sense of Ryszard Deszcz, in: Geometry

and Topology of Submanifolds, VI, World Sci., River Edge, NJ, 1994, 199-209.

On Some Applications of Invariants of Continuous Subgroups of thePoincare Group P (1, 4) in Mathematical Physics

Vasyl Fedorchuk(Institute of Mathematics, Pedagogical University,Krakow, Poland;

Pidstryhach Institute of Applied Problems of Mechanics andMathematics of NAS of Ukraine)

Volodymyr Fedorchuk(Franko Lviv National University,Lviv, Ukraine)

Using functional bases of invariants of continuous subgroups of the Poincare groupP (1, 4) we have described non–singular manifolds (in the spaces M(1, 3)×R(u) andM(1, 4) × R(u)) which are invariant under these subgroups. R(u) is the numberaxis of the dependent variable u. These manifolds have been used for symmetryreduction of some differential equations which are important in theoretical andmathematical physics.

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Besides, taking into account functional bases of the first–order differential in-variants of continuous subgroups of the group P (1, 4) we have constructed non–singular differential invariant manifolds (in the spaces M(1, 3) × R(u)(1) andM(1, 4) × R(u)(1)) which are invariant under these subgroups. R(u)(1) is the firstprolongation of R(u). Every such manifold is a class of the first– order differen-tial equations (in the spaces M(1, 3)× R(u) or M(1, 4)× R(u)) with non–trivialsymmetry group.

Loops which are semidirect products of groupsAgota Figula (University of Debrecen)

In this talk I shall show that a wide class of proper loops L can be representedwithin the group of affinities of an affine space A of dimension 2n over a com-mutative field K. They are semidirect products of groups of translations of A bysuitable subgroups of GL(2n, K). For many of them we may take as elements affinen-dimensional transversal subspaces of A. This representation of the loops L de-pends in an essential manner on the existence of a regular orbit in the set of affinen-dimensional transversal subspaces of A of for the group generated by the lefttranslations of L.

To realize our examples it is important to know the eigenvalues for certain subsetsof products of matrices in GL(n, K).

In particular we obtain in this manner smooth loops having Lie groups of affinereal transformations as the groups generated by the left translations, whereas thegroups generated by the right translations are smooth groups of infinite dimension.The Akivis algebras of these loops are also studied; these are semidirect products ofLie algebras. Moreover I show, that there are non-connected proper smooth loopshaving Lie algebras as their Akivis algebras.

The Laplace spectrum and Hermitian manifoldsLew Friedland (State University of New York (Geneseo), US)

The spectrum Specp (Mn, g) = λi,p of eigenvalues of the Laplace operator ondifferential p-forms 0 ≤ p ≤ n on compact Riemann and Kahler manifolds (Mn, g)determines geometric properties of the manifold, though isospectral manifolds arenot necessarily isometric, even locally. We consider here some consequences ofisospectrality on almost Hermitian manifolds where some restriction on the curva-ture or type of curvature pinching is assumed and use classical spectral methods toderive geometric results.

Stability and canonical Kahler metricsAkito Futaki (Tokyo Institute of Technology)

In this talk I will review a conjecture on K-stability and the existence problem ofextremal Kahler metrics. K-stability measures the stability of degenerations of thepolarized manifold in consideration, and this stability condition is conjectured tobe equivalent to the existence of Kahler metric of constant scalar curvature.

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Acceleration bundles of infinite dimensional manifolds and applicationsG. Galanis (Naval Academy of Greece-Section of Mathematics),E. Vassiliou (University of Athens-Department of Mathematics),C.T.J. Dodson (Manchester University-School of Mathematics)

The second order (acceleration) tangent bundle T 2M of an infinite dimensionalmanifold M , consisting of all equivalent classes of curves on M that agree up totheir second derivative, is always a fibre bundle over M . However, the definitionof a vector bundle structure on T 2M requires the existence of a linear connection∇ on the base, in case of Banach modeled manifolds, as well as additional proper-ties for the corresponding Christoffel symbols in the framework of Frechet smoothmanifolds.

The obtained vector bundle structure on T 2M remains invariant if ∇ is replacedby a conjugate connection with respect to any diffeomorphism of M . Furthermore,this structure is associated with the principal bundle of second order linear framesand provides a convenient framework for the study of second order differential equa-tions on the base manifold M with several potential applications in MathematicalPhysics.

Complex extensions of semisimple symmetric spacesLaura Geatti (Universita di Roma)

Let G/H be a pseudo-Riemannian semisimple symmetric space. The tangent bun-dle T (G/H) contains a maximal G-invariant neighbourhood of the zero sectionwhere the adapted complex structure (in the sense of Lempert-Szoke) exists. Sucha neighbourhood is endowed with a canonical G-invariant pseudo-Kaehler metricof the same signature as the metric on G/H. One can exploit the polar mapφ : T (G/H) → GC/HC and its properties to push-forward such a metric ontodistinguished G-invariant domains in GC/HC or onto coverings of principal orbitstrata in GC/HC .

Taut representations of compact simple Lie groupsClaudio Gorodski (University of Sao Paulo)

We classify the representations of the compact simple Lie groups all of whose orbitsare tautly embedded in Euclidean space with respect to Z2-coefficients.

Simply connected two-step homogeneous nilmanifolds of dimension 5Szilvia Homolya (Institute of Mathematics, University of Miskolc)

A connected Riemannian manifold which admits a transitive nilpotent Lie groupN of isometries is called a nilmanifold. Two-step homogeneous nilmanifolds arestudied intensively in the last twenty years. The aim of the talk is to classify allsimply connected two-step nilpotent Lie groups of dimension 5 equipped with left-invariant metrics up to isometry. We also calculate the corresponding isometrygroups.

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Finsler Metrics of Constant Flag CurvatureDragos Hrimiuc (Department of Mathematics, University of Alberta)

We will consider the transformation of the flag curvature tensor and examine theinvariance of constant flag curvature Finsler metrics through affine deformations.

Stability numbers in K-contact manifoldsAna Hurtado (Departamento Geometria y Topologia, Universidad de Valencia)We study the stability of the characteristic vector field of a K-contact manifoldas a critical point of the energy and volume functionals, when we consider a twoparametric variation of the metric. First of all, we multiply the metric in thedirection of the characteristic vector field by a constant and then we change thenew metric by homotheties. We will see that the results obtained in [1] for Sasakianmanifolds are valid for a K-contact manifold not necessarily Sasakian. Finally wewill apply these results to study the stability of Hopf vector fields on Berger sphereswhen we change the Berger metrics by homotheties.

References

[1] V. Borrelli, Stability of the characteristic vector field of a Sasakian Manifold, Soochow J. of

Math. 30, 283-292, (2004).

Higher order Utiyama’s invariant interactionJosef Janyska (Masaryk University, Brno)

We prove higher order version of the Utiyama’s invariant interaction of a particlefield and a gauge field (a principal connection on a principal bundle). To describethe Utiyama’s interaction in order r ≥ 2 we have to use an auxiliary linear symmet-ric connection on the base manifold. We prove that any natural (invariant) operatorof order s in the linear symmetric connection on the base manifold, of order r inthe gauge field and of order k in the particle field, s, r ≥ k − 1, s ≥ r − 2, withvalues in a (1, 0)-order G-gauge-natural bundle factorizes through the curvaturetensors of both connections, the particle field and their covariant differentials up tosufficiently high orders. The covariant differentials are considered with respect toboth connections.

Remarks on almost contact metric structuresToyoko Kashiwada (Meiji University, Japan)

Some classes of almost contact metric manifolds are connected with locally con-formal almost Kahler manifolds (l.c.a. K.-manifolds) as product manifolds with aunit circle. In this talk, we study almost contact metric structures from such astandpoint.

We introduce some classes of almost contact metric structures, and discuss theircurvature properties. We further characterize them in connection with the cosym-pletic structure.

We also introduce some kinds of curvature operators acting on subspaces of2-forms and discuss them in our manifolds.

(An almost Hermitian structure is called an l.c.a.K.-structure if its fundamental2-form Ω satisfies dΩ = 2α ∧ Ω with a closed 1-form α. )

13

A generalization of Cartan hypersurfacesMakoto Kimura (Department of Mathematics, Shimane University)

We call isoparametric minimal hypersurfaces with 3 distinct principal curvaturesin a sphere as Cartan hypersurfaces. Such hypersurfaces are austere submanifoldswhich were defined by Harvey-Lawson, and they showed that the twisted normalcone of austere submanifolds in a sphere is a special Lagrangian cone.

We will show, as a generalization of 3-dimensional Cartan hypersurface, thatfrom first order isotropic complex submanifolds Σ in a complex quadric, we canconstruct austere immersion form some circle bundle over Σ to a sphere. Holomor-phic curves in a complex quadric are first studied by Jensen-Rigoli-Yang.

On the functorial prolongations of principal bundlesAntonella Kabras (University of Florence),

Ivan Kolar (Department of algebra and geometry, Masaryk University)We deduce several general properties of the functorial prolongations of principalfiber bundles with respect to a fiber product preserving bundle functor F on thecategory of fibered manifolds with m-dimensional bases and fiber preserving mapswith local diffeomorphisms as base maps. We clarify that this theory is based onthe concept of weak principal bundle, in which the structure group is replaced by agroup bundle. Then we describe the related Lie algebroids in terms of the actionsof a Lie algebroid on a Lie algebra bundle. Our approach is essentially based onour previous characterization of the functor F by means of Weil abgebras.

Hodge type decompositionWojciech Kozlowski (Polish Academy of Sciences, Lodz Branch)

In the space Λp of polynomial p-forms in Rn we introduce some special inner prod-uct. Let Hp be the space of polynomial p-forms which are both closed and co-closed.We prove that Λp splits as the direct sum d?(Λp+1)⊕δ?(Λp−1)⊕Hp, where d? and δ?

denote the adjoint operators to d and δ with respect to that special inner product.

Weinstein’s theorem for Finsler manifoldsLaszlo Kozma (University of Debrecen)

Ion Radu Peter (Technical University of Cluj-Napoca)Here we prove the generalization of Weinstein’s theorem for Finsler manifolds: anisometry of a compact oriented Finsler manifold of positive sectional curvaturehas a fixed point supposed that it preserves the orientation of the manifold if itsdimension is even, or reverses it if odd. Some consequences will be also analysed.

14

Canonical symplectic structures on the r-th order tangent bundles of asymplectic manifold

J. Kurek (Maria Curie Sklodowska University, Lublin)W.M. Mikulski (Jagiellonian University, Krakow)

We describe all canonical symplectic structures Λ(ω) on the r-th order tangentbundle T rM = Jr

0 (R,M) of a symplectic manifold (M,ω). More precisely, wededuce that Λ(ω) is a linear combination of Morimoto’s k-lifts of ω) with respectivereal coefficients. Some similar results are remarked, too.

Stochastic Poisson-Sigma modelR. Leandre (Universite de Bourgogne)

Cattaneo-Felder have defined a path integral representation of Kontsevitch for-mula of a ∗-product. Klauder have defined stochastic regulator of Hamiltonianpath integrals. We define a field theoretical analoguous of Klauder’s formalism forthe stochastic Poisson-Sigma model of Kontsevitch, by using infinite dimensionalBrownian motion of Airault-Malliavin. We perform the semi-classical limit.

Bounding from below the degree of an algebraic differential systemhaving a prescribed algebraic solution

D. Lehmann and V. Cavalier(Universite des Sciences et Techniques de Montpellier)

Let d be the degree of an algebraic one-dimensional foliation F on the complexprojective space Pn. Let Γ be an algebraic solution of degree δ, and geometricalgenus g. We wish to bound d from below in function of Γ. We shall improve thepresently known results on the subject, in the following way.

The normal bundle NΓ0 to the non-singular part Γ0 of Γ in M has a stable classwhich always admits a natural extension [NΓ] in the Grothendieck group K0(Γ).[If Γ is moreover a locally complete intersection (LCI) in M , [NΓ] may even berealized as the stable class of a natural bundle NΓ which is a natural extension ofNΓ0 to all of Γ ]. Denote by Σ = SingΓ ∪ (SingF ∩ Γ) the union (made of isolatedpoints) of the singular part of Γ with the set of singular points of F which are inΓ. To each point mα in Σ, we can associate an integer GSV mα(F ,Γ) , such thatwe get:Theorem(i) The following formula holds:

(d + n)δ −(c1([NΓ]) _ [Γ]

)=

∑α

GSV mα(F ,Γ).

(ii) The following inequality holds:∑α

GSV mα(F ,Γ) ≥ B(Γ)− E(Γ),

where B(Γ) denotes the total number of locally irreducible branches through singularpoints of Γ when Γ has singularities, and B(Γ) = 1 (instead of 0) when Γ is smooth,

and E(Γ) = 2− 2g + c1

([NΓ]

)_ [Γ]− (n + 1)δ denotes the correction term in the

genus formula (g being the geometrical genus of Γ) .

15

When Γ =⋂n−1

λ=1 Sλ is the complete intersection (not necessarily smooth) of n−1algebraic hyper-surfaces Sλ in Pn of respective degree δλ (1 ≤ λ ≤ n − 1), we getthe following inequality: (

d + n−n−1∑λ=1

δλ

)δ ≥ B(Γ)− E(Γ),

and in particular: (d + 2− δ)δ ≥ B(Γ)− E(Γ) for n = 2.

These results are also refined when Γ is reducible. Examples of applications willbe given.

Stationary surfaces in Lorentz-Minkwoski spaceRafael Lopez (Universidad de Granada, Spain)

In Lorentz-Minkowski space L3, we consider the following variational problem. LetΠ be a spacelike plane and denote by Π+ one of the two halfspaces at which Πdivides L3. Let M be a compact spacelike surface whose boundary ∂M lies in Πand int(M) ⊂ Π+ and let us denote Ω the bounded domain by ∂M in Π. Weconsider all perturbations in such way that M is adhered to Π (∂M ⊂ Π) andint(M) remains in Π+. We consider the following energy functional

E = |M | − coshβ|Ω|+∫

M

Y dM,

where |M | and |Ω| denote the area of M and Ω respectively. Here Y is a potentialthat, up constants, measures at each point the distance to Π. We are interestedin those configurations in a state of equilibrium, that is, when the energy of thesystem is critical under any perturbation that do not change the volume enclosedby M ∪Ω. According to the principle of virtual work, the equilibrium of the systemis achieved if the following two conditions hold: i) the mean curvature H of M is alinear function on the distance to Π: H(x) = κx3(x)+λ, κ indicates the capillarityconstant and λ is a Lagrange multiplier arising from the volume constraint and ;ii) the hyperbolic angle β with which M and Π intersect along ∂M is constant.In such case, we shall say that M is a stationary hypersurface. In absence of thetimelike potential Y , M is a surface with constant mean curvature. The first resultis

Theorem 1 Given a stationary surface in L3, there is a vertical straight-lineL orthogonal to Π about which M is rotational symmetric. Moreover, M is topo-logically a disc and the intersection of M with a plane orthogonal to L is a circlewhose center lies on the axis L.

Assuming then rotational symmetry, the Euler equation becomes an ordinarydifferential equation for the profile curve that defines the stationary surface. Wediscuss in some detail the solutions of this differential equation starting for theproblem of existence and uniqueness. As consequence,

Theorem 2. Given κ, β and Π, there exists a stationary surface supported on Πand where β is the angle of contact between the surface and Π. The drop is uniqueup isometries of the ambient space L3.

Depending on the sign of κ, there exists a qualitative difference of shapes thatadopt a stationary surface. We analyze the shapes that a stationary surface canadopt by deriving estimates of the height, volume and properties of monotonicity.For example, with the above notations,

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Theorem 3. If κ > 0, for any stationary surface M resting on Π where R isthe radius of the disc of Π that wets M , the height q of M satisfies:

q < Rcoshβ − 1

sinhβ.

Finally, we point up also an existence result in terms of the volumeTheorem 4. For each κ, β and a positive real number V, there exists a unique

stationary surface resting on Π enclosing a liquid of volume V and that makes aconstant angle β with Π along ∂M .

References

[1] R. Lopez, Spacelike hypersurfaces with free boundary in Minkowski space under the effect ofa timelike potential, preprint.

[2] R. Lopez, Stationary liquid drops in Lorentz-Minkowski space, preprint.

[3] R. Lopez, An exterior boundary value problem in Minkowski space, preprint.

For Symmetries of Special ManifoldsM. Belkhelfa (C.U.- Mascara, Algerie),

K. Matsumoto (Yamagata University),L. Verstraelen (Catholic University, Leuven)

We know many kinds of the notion of symmetries (locally symmetry, Weyl semi-symmetry, pseudo-symmetry, etc.) of (semi) Riemannian manifolds and they havea lot of interesting properties.

In the present talk, we mainly show that a Weyl semi-symmetric Kenmotsu man-ifold is a space form with constant curvature −1. Then we define a new symmetry(named phi-pseudo-symmetry) of a semi Riemannian manifold. In particular, weget some properties of this symmetry in Sasakian, Kenmotsu and LP -Sasakianmanifolds.

Submanifolds with nonzero mean curvature in a Euclidean sphereYasushi Uchida and Yoshio Matsuyama

(Department of Mathematics, Chuo University, Japan)The purpose of this paper is to prove the following theorem:Theorem. Let Mn be a complete, connected and orientable submanifold withnonzero constant mean curvature H in Sn+2(c). Moreover, we put |φ|2 = S − nH2

and BH the square of the positive root of the equation x2 +n(n− 2)√n(n− 1)

Hx−n(H2 +

c) = 0 with respect to x, where S denotes the squared norm of the second funda-mental form. If |φ|2 satisfies

|φ|2 ≤ BH for all x ∈ Mn,

then Mn lies in a totally geodesic hypersurface Sn+1(c) of Sn+2(c) and(1) either |φ|2 ≡ 0 and Mn is totally umbilic or |φ|2 ≡ BH .(2) |φ|2 ≡ BH if and only if

(B) n ≥ 3 and Mn = Sn−1(r1) × S1(r2) ⊂ Sn+1(c) where r21 + r2

2 =1c

and

r21 <

n− 1ncor

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(C) n = 2 and M2 = S1(r1)× S1(r2) ⊂ S3(c) where r21 + r2

2 =1c

and r21 6=

12c

.

The following generalized maximum principle due to Omori and Yau will be usedin order to prove our theorem.Generalized Maximum Principle (Omori and Yau). Let Mn be a completeRiemannian manifold whose Ricci curvature is bounded from below and f ∈ C2(M)a function bounded from above on Mn. Then, for any ε > 0, there exists a pointp ∈ Mn such that

f(p) ≥ sup f − ε, ||grad f || < ε, ∆f(p) < ε.

Non-holonomic reduction by stagesTom Mestdag (Department of Mathematical Physics and Astronomy, Ghent

University)In this talk we study reduction of non-holonomic systems with symmetry. Based onLie algebroid theory, we develop a geometric context in which successive reductioncan be described and performed. We prove, among other, that reduction in twostages is equivalent with direct reduction.

Locally strongly convex surfaces with complete flat affine metricA. Martınez (Universidad de Granada), F. Milan (Universidad de Granada) and

L. Vrancken (Universite de Valenciennes)We study locally strongly convex surfaces with complete Blaschke metric. We showhow we can characterize all known examples by a tensorial condition involving thecovariant derivative of the shape operator and the gradient of the Pick invariant.

Special transformations with projective motion in projective Finslerspace

Chayan Kumar Mishra (Dr. R. M. L. Avadh University, Faizabad),The concept of projective Finsler space (P-Fn) have been studied by R. B. Misraand F. M. Meher [Tensor, N.S. Vol-(23) , (1972), 57-65]. Projective motion de-fined by contra, concurrent, spacial concircular transformations and torse-formingtransformation (which can be viewed as special cases above transformations) in arecurrent Finsler space and Finsler space where discussed in detail by R. B. Misra[J. South Gujrat University, 6, (1977),72-96], [Boll.Un. Mat. Ital. (5), 16B (1979),32-53] and R. B. Misra and C. K. Mishra [Tensor, N. S. , Vol.65 (2004), 1-7] respec-tively. Presently we study the projective motion defined by the existence of a specialtransformations i.e. contra transformation, concurrent transformation, spacial con-circular transformation, recurrent transformation, concircular transformation andtorse-forming transformation in a projective Finsler space.

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On the isometry groups of the 8 homogeneous 3-spaces (Classificationsand projective metric presentations)

Emil Molnar (Budapest University of Technology and Economics, Dept. ofGeometry),

The 8 homogeneous Thurston geometries E3, S3, H3, S2×R, H2×R,SL2×R, Nil,Sol can be interpreted in the projective-metric 3-sphere PS3(V 4, V 4, R, 〈, 〉), wherea projective polarity or scalar product 〈, 〉 in V 4 is distinguished [3]. This describesthe orthogonality of 2-planes, as usual, in the sense of F. Klein’s Erlangen-Program.The signature of this scalar product, together with some invariant elements giveus the possibility to characterize the compact and other orbifolds of these spacesexplicitely.

Thus, some concrete examples and counter-examples will be given, maybe sur-prisingly (to [1], [2], [4], [5]), in Nil, H2 ×R and SL2 ×R, respectively.

References

[1] W.D. Dunbar, Geometric orbifolds, Rev. Math Univ. Compl. Madrid, 1 (1988), 67-991.[2] E. Molnar, Discontinuous groups in homogeneous Riemannian spaces by classification of

D-symbols, Publ. Math. Debrecen, 49/3-4 (1996), 265-294.

[3] E. Molnar, The projective interpretation of the eight 3-dimensional homogeneous geometries,Beitrage zur Algebra und Geometrie, Vol. 38 No. 2, (1997), 261-288.

[4] E. Molnar, I. Prok, and J. Szirmai, Classification of tile-transitive 3-simplex tilings and theirrealizations in homogeneous spaces, in: Non-Euclidean Geometries, Janos Bolyai Memorial

Volume, Ed. A. Prekopa and E. Molnar, Kluwer - Springer (2005), pp. 325-367 (to appear).

[5] W. Thurston (and S. Levy editor), Three-dimensional geometry and topology, Vol. 1, Prince-ton University Press, Princeton, New Jersey (1997) (Ch. 4.7).

Cartan prolongations of distributions and monster manifoldsPiotr Mormul (Warsaw University, Warsaw, Poland)

In the years 90 of the XX century, through the works of (mainly) Bryant, Hsu,Jean, Montgomery and Zhitomirskii, it had become clear that the geometric dis-tributions generating 1-flags, often called Goursat, were locally nothing more thanthe outcomes of several Cartan prolongations started from the tangent bundle toa 2-surface. This helped to better understand Goursat objects and, among others,to advance the local classification of them. (Author’s talk at the Debrecen 2000’conference was surveying those developments.)

In the first years of the present century, through the works of (mainly) Kumpera,Rubin, and the author, it has become clear that the geometric distributions gener-ating special k-flags, k ≥ 2, are locally nothing more than the outcomes of severalgeneralized Cartan prolongations started from the tangent bundle to a (k + 1)-manifold. This decisively helps to produce local polynomial pseudo-normal formsfor special k-flags and to start the local classification of them.

Thanks to Cartan prolongations, both 1-flags and special k-flags, k ≥ 2, appearto possess, in any fixed length r, universal models living on huge ‘monster’ man-ifolds, those maifolds having utmostly simple constructions. That is to say, themonster manifolds in length r carry certain super Goursat- or special multi-flagsthat locally model any Goursat- or special multi-flag existing in that length r. Wewould like to report on these constructions during the Conference.

19

The equations of a holomorphic subspace in a complex Finsler spaceGheorghe Munteanu (Transilvania University of Brasov, Romania )

In a previous paper we stippled the motifs of holomorphic subspace of a complexFinsler space: induced nonlinear connection, coupling connections and the inducedtangent and normal connections. In the present paper we investigate the equationsof Gauss, H− and A−Codazzi, Ricci equations of a holomorphic subspace. Isdeduced the link between the holomorphic curvatures of Chern-Finsler connectionand its induced tangent connection. A condition for totally geodesic holomorphicsubspace is obtain.

Hyper-Kahler extensions of complex-symplectic structures oncotangent bundlesIhor V. Mykytyuk

(Institute of Mathematics, University of Rzeszow, Poland andInstitute of Applied Problems of Mathematics and Mechanics, Lviv, Ukraine)

Let G/K be an irreducible Hermitian symmetric space of compact type with thestandard homogeneous complex structure. Then the real symplectic manifold(T ∗(G/K),Ω) has the natural complex structure J−. Moreover, the manifold(T ∗(G/K),Ω + iΩ′), where Ω′(·, ·) = Ω(−J−·, ·), is a complex-symplectic mani-fold with respect to the complex structure J−. It is well known that there existsa unique G-invariant hyper-Kahler extension (g, J1, J2) of this complex-symplecticstructure such that the restriction of the metric g to G/K coincide with the homo-geneous Kahler metric on G/K. For this extension J1 = J− and ω2 = Ω, ω3 = Ω′,where ω2 and ω3 are the Kahler forms of the Kahler structures (g, J2) and (g, J3)(J3 = J1J2 = −J2J1) respectively.

But G/K is a homogeneous symplectic manifold with (Kahler) 2-form ω. So it isnatural to consider the complex-symplectic structures ω2 + iω3 on T ∗(G/K) of theform aΩ + bΩ′ + cω, a, b, c ∈ C, c 6= 0. We prove that there is a 5-dimesional (real)family of such structures admitting G-invariant hyper-Kahler extension defined onthe whole cotangent bundle T ∗(G/K). In the simplest case of the two-dimensionalsphere G/K = SO(3)/SO(2) we prove that some of the corresponding hyper-Kahlermetrics are complete. These metrics generalize well known Eguchi-Hanson metricson T ∗S2.

Isomorphism of differentiable loops on the real linePeter T. Nagy (Institute of Mathematics, Debrecen University)

The paper is devoted to the study of differentiable proper loops L on the real linesuch that the group G topologically generated by the left translations is locallycompact and hence it is isomorphic to the universal covering group of PSL2(R).Using the methods developed in [1] there is given a class of natural parametrizationof the loop manifold L corresponding to the Iwasava decompositions of G and char-acterize the differentiable curves R → G on the manifold G, the points of which canbe interpreted as the left translations of the loop L. The obtained parametrizationyields a generalized global canonical coordinate system on the differentiable loopdefined on R, the loop multiplication with respect to this parametrization can be

20

expressed by

x · y =

x + arccot(2ν(x) + e−2δ(x) cot y

)+ iπ, if iπ < y < (i + 1)π,

x + iπ, if y = iπ,

for any i ∈ Z, where δ(t) and ν(t) are arbitrary differentiable functions on Rsatisfying the initial conditions δ(0) = ν(0) = 0, δ(0) = dδ

dx (0) = 0, ν(0) = dνdx (0) = 0

and the differential inequality

δ(t)2 +12ν(t)− 1 < ν(t)δ(t).

Using this representation we give necessary and sufficient conditions for the isomor-phism of differentiable loops on R such that the group G topologically generatedby the left translations is locally compact.

References

[1] P. T. Nagy–K. Strambach, Loops in Group Theory and Lie Theory, Expositions in Mathe-

matics 35, Walter de Gruyter, Berlin–New York (2002).

[2] P. T. Nagy–K. Strambach, Coverings of topological loops, manuscript (2004).[3] P. T. Nagy–I. Stuhl, Differentiable loops on the real line, manuscript (2005).

Quest for relations in Diff(C, 0)Isao Nakai (Ochanomizu University)

A germ of holomorphic diffeomorphism of C fixing 0 is formally conjugate to atime-1 map of a holomorphic vector field on C vanishing at 0, or a composite oftwo time-1 maps. Therefore a word of germs of holomorphic diffeomorphisms is acomposite of some time-1 maps of formal vector fields. First we give a formula tocalculate the Taylor coefficients of time-1 transport maps of formal ordinary dif-ferential equations. Then we apply the formula to the differential equation definedby a word of time-1 maps to seek relations in the group of germs of holomorphicdiffeomorphisms Diff(C, 0). We give the various results on the existence of relationsof formal diffeomorphisms.

Curvature homogeneous 3-dimensional Lorentz ManifoldsS. Nikcevic (Mathematical Institute, SANU, Belgrade)

P. Gilkey (University of Oregon, Eugene, USA)We study a family of 3-dimensional Lorentz manifolds. Some members of thefamily are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvaturehomogeneous. All are 0-modeled on indecomposible local symmetric spaces. Someof the members of the family are geodesically complete, others are not. All havevanishing scalar invariants.

21

Masless particles in warped three spacesMiguel Ortega (Universidad de Granada)

The model governed by an action measuring the total proper acceleration of trajec-tories provides a nice one to describe the dynamics of massless relativistic particles.In high rigidity cases, metrics with constant curvature, the model is consistent onlyin spherical three spaces and in three dimensional anti de Sitter backgrounds, ac-cording to a Riemannian or a Lorentzian context, respectively. In contrast with flatgravitational fields, the existence of non trivial trajectories are shown in a familyof three spaces whose metrics admit a certain degree of symmetry, being such tra-jectories included in regions with real presence of matter. An algorithm to obtainthem is also designed.

Constraints in gauge-natural invariant field theoriesM. Palese and E. Winterroth

(University of Torino)We specialize in a new way the Second Noether Theorem for gauge-natural fieldtheories by relating it to the Jacobi morphism and show that it plays a funda-mental role in the derivation of canonical covariant conserved quantities. Recallthat generalized Bergmann–Bianchi identities in gauge-natural field theories arein fact necessary and (locally) sufficient conditions for a Noether conserved cur-rent to be not only closed but also the divergence of a skew-symmetric (tensor)density - a superpotential - along solutions of the Euler–Lagrange equations. Inparticular, we show that Bergmann–Bianchi identities in such field theories holdtrue covariantly and canonically only along solutions of generalized gauge-naturalJacobi equations. Vice versa all vertical parts of gauge-natural lifts of infinitesi-mal principal automorphisms lying in the kernel of generalized Jacobi morphismssatisfy Bergmann–Bianchi identities and thus are generators of canonical covariantcurrents and superpotentials. Thus, generalized Bergmann–Bianchi identities holdtrue in a canonical covariant way if and only if the second variational derivative -with respect to vertical parts of gauge-natural lifts - of the Lagrangian vanishes. Wefurther show that the indeterminacy appearing in the derivation of gauge-naturalconserved charges - i.e. the difficulty of relating in a natural way infinitesimal gaugetransformations with infinitesimal transformations of the basis manifold - can besolved by using this constraint: for any gauge-natural invariant field theory we findthat the above mentioned indeterminacy can be always solved canonically.

Spectral Geometry of Riemannian V-submersionsJeong Hyeong Park (Honam University)

We study when the pull-back of an eigenform of the Laplacian on the base of acompact Riemannian V -submersion is an eigenform of the Laplacian on the totalspace of the submersion, and when the associated eigenvalue can change.

22

Natural equivariant quantizations by means of Cartan connectionsFabian Radoux (University of Liege)

In the framework of geometric quantization, one can think of a quantization proce-dure as a linear bijection from the space of classical observables (also called symbols)to a space of differential operators acting on wave functions.

It is known that there is no natural quantization procedure. In other words, thespaces of symbols and of differential operators are not isomorphic as representa-tions of Diff(M). However, when there is a Lie group G acting on M by localdiffeomorphisms, P. Lecomte and V. Ovsienko defined a G-equivariant quantizationas a linear bijection from the space of symbols to the space of differential operatorsthat exchanges the actions of G on these spaces.

They considered the case of the projective group PGL(m + 1, R) acting on themanifold M = Rm by linear fractional transformations. This leads to the notionof projectively equivariant quantization. One of the main results in this case is theexistence of a projectively equivariant quantization and its uniqueness, up to somenatural normalization condition. The authors also showed that their results couldbe directly generalized to the case of a manifold endowed with a flat projectivestructure.

Together with C. Duval, they then considered the conformal group SO(p+1, q+1) acting on the space R(p+q) or on a manifold endowed with a flat conformalstructure. There again, they obtained results of existence and uniqueness of aconformally equivariant quantization, up to normalization. At that point, all theresults were dealing with a manifold endowed with a flat structure. Recently, P.Lecomte conjectured the existence of a quantization procedure depending on atorsion-free connection, that would be natural (in all arguments) and that wouldbe left invariant by a projective change of connection. The existence of such aquantization procedure was proved by M. Bordemann, using the notion of Thomas-Whitehead connections.

We will show that the existence of a natural and projectively equivariant quanti-zation map can be obtained using Cartan connections and we will derive an explicitformula for the quantization map in terms of these Cartan connections. We willalso show how the methods extend to deal with the conformal case.

Lipschitz vs. smooth homogeneityDusan Repovs (University of Ljubljana)

Smooth: Repovs, Skopenkov and Scepin have proved that a locally compact (pos-sibly nonclosed) subset of Rn is smoothly homogeneous if and only if it is a smoothsubmanifold of Rn. Lipschitz: Garity, Repovs and Zeljko have constructed Lips-chitz homogeneous wild sets in R3. So Lipschitz homogeneity cannot detect mani-folds in this category. We shall explain how this work is related to our work on theclassical Hilbert–Smith Conjecture.

Remark: The Bing-Borsuk Conjecture asserts that finite dimensional (topolog-ically) homogeneous ANR’s are (topological) manifolds. It is true for dimensionsbelow 3, whereas in dimension 3 it implies the Poincare Conjecture. In higherdimensions counterexamples have been announced.

23

Stationary sets and constant mean curvature surfaces in theHeisenberg group

Cesar Rosales (University of Granada)We study sets of the n-dimensional Heisenberg group Hn which are critical points,under a volume constraint, of the sub-Riemannian perimeter associated to the dis-tribution of horizontal vector fields in Hn. This leads to a notion of mean curvaturefor hypersurfaces, and to the development of a theory of constant mean curvature(CMC) hypersurfaces in this sub- Riemannian setting.

In this talk we gather some classification results on CMC surfaces in H1, andwe show new examples of non-zero CMC hypersurfaces in Hn. We also provideapplications of all this to the isoperimetric problem in Hn.

On topology of saddle submanifoldsVladimir Rovenski (University of Haifa)

The classes of submanifolds: (k, ε)-saddle, (k, ε)-parabolic, and (k, ε)-convex areintroduced in terms of eigenvalues of their second fundamental form. These classesfor ε > 0 generalize known classes of k-saddle, etc. submanifolds, and naturallyarise among submanifolds with extrinsic curvature bounded from above and smallcodimension. The relations between curvature and topology of these submanifoldsare studied.

On fragmentations of diffeomorphisms on a manifoldTomasz Rybicki and Jacek Lech

(Faculty of Applied Mathematics, AGH University of Science and Technology,Cracow, Poland)

Let G(M) ⊂ Diffr(M) be a diffeomorphism group, and Gc(M)0 be its compactlysupported identity component. Suppose that for an isotopy ft in Gc(M)0 withsupp(ft) ⊂

⋃ki=1 Ui, Ui are open, there exist isotopies fj,t in Gc(M)0, j = 1, . . . , l,

with ft = f1,t . . . fl,t such that supp(fj,t) ⊂ Ui(j) for all j. Then G(M) is saidto satisfy a fragmentation property of the first kind.

Our idea is to introduce also the second kind of fragmentations. Such frag-mentations are considered only for isotopies and diffeomorphisms in a sufficientlysmall C1 neighborhood of the identity in the group Gc(Rn)0. On the other hand,we claim that the factors of the fragmentation are uniquely defined by the initialdiffeomorphism and that the Cr-norms of the factors are controlled in a convenientway by the Cr-norms of the initial diffeomorphism.

In proofs of some theorems on the simplicity and perfectness of diffeomorphismgroups a clue role is played by both kinds of fragmentation properties. Here weshow a new perfectness theorem on groups of diffeomorphisms preserving specialdomains, where both kinds of fragmentations are applied.

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Theory of connections and problem of existence of Backlundtransformations for general second-order partial differential equations

A.K.Rybnikov1 ((Moscow State M.V. Lomonosov University, Russia))The geometrical theory of Backlund transformations (BT) for general second-orderpartial differential equations

Φ(xi, z, zj , zkl) = 0 (i, j, · · · = 1, . . . , n) (1)

is developed here as a special chapter in the theory of connections. We associatethe Backlund maps (and, in particular, the BT) with the special connections [1]defining the representations of zero curvature for (1). Formerly [2] we presented ananalogous theory for the evolution equations

zt − f(t, xi, z, zj , zkl) = 0. (2)

BT for the evolution equations (singular BT) are more special than that for generalPDE (general or non-singular BT). The problem of existence of BT for general PDEis more difficult than an analogous problem for evolution equations.

In this work we state that one can represent the Backlund system (e.g. a partialdifferential system defining the Backlund map) for a second-order equation (1) ofgeneral type in case n = 2 in the following form

y1 = −γ211(x

i, z, zj) + y · γ1(xi, z, zj) + y2 · γ121(x

i, z, zj),y2 = −γ2

12(xi, z, zj) + y · γ2(xi, z, zj) + y2 · γ1

22(xi, z, zj)

, (3)

where γ1, γ211, γ1

21, γ2, γ212, γ1

22 are coefficients of a special connection defining therepresentation of zero curvature for a given equation (1). For an evolution equation(2) with one space variable the Backlund system is of another form [2]

yt = −y · g101(t, x, z, zx) + γ1

00(t, x, z, zx),yx = −y · γ1

11(t, x, z, zx) + γ101(t, x, z, zx)

, (4)

where g101, γ1

00, γ101, γ1

11 are coefficients of a special connection defining the repre-sentation of zero curvature for a given evolution equation.

We prove that any second-order differential equation (1), where n = 2, admittinggeneral Backlund maps is equivalent to some differential equation of the followingform

K(xi, z, zj) · z11 + L(xi, z, zj) · z12 + M(xi, z, zj) · z22 + H(xi, z, zj) = 0 (5)

Also a number of necessary conditions for existence of standard BT (for which theconnection coefficients depend on z, z1, z2 only and, in addition, γ2 = γ2(z), γ2

12 =γ212(z), γ1

22 = γ122(z) and the derivatives of γ2, γ2

12, γ122 do not vanish simultaneously)

are found. It is shown that any equation (1), where n = 2, which admits non-singular standard BT, is equivalent to some differential equation of the form

F1(z, z1, z2) · z12 + F2(z, z1, z2) · z22 + H(z, z1, z2) = 0. (6)

Any quasilinear parabolic equation of the form

z22 −H(z, z1, z2) = 0 (7)

is the equation of type (6) but not every of them admits BT. We prove that if theequation (7) admits non-singular standard BT then it is of the form

z22 − F (z, z2) · z1 −G(z, z2) = 0. (8)

1Supported by RFFI, grant 05-01-01015A

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In particular case, when F (z.z2) depends only on z and does not vanish, the equa-tion (8) may be represented in the following way

z1 − f(z) · z22 − g(z, z2) = 0 (f(z) 6= 0). (9)

Also not every equation of this type admits BT. We prove that if tthe equation (9)admits standard non-singular BT then it is of the form

z1 − f(z) · z22 − ξ(z) · (z2)2 − η(z) · z2 − ζ(z) = 0 (f(z) 6= 0). (10)

Moreover, if ζ(z) 6= 0, then

ζ(z) =f(z)fξ(z)

(A

∫η(z) · fξ(z)

f(z)dz + B

∫fξ(z)f(z)

dz −A2

∫fξ(z)dz + C

), (11)

where A,B,C = const (A2 + B2 + C2 6= 0), fξ(z) = eR ξ(z)

f(z) dz. Here∫ ξ(z)

f(z)dz,∫fξ(z)dz,

∫ fξ(z)f(z) dz,

∫ η(z)·fξ(z)f(z) dz are the arbitrary chosen antiderivatives.

In the last section of this work it is proved the existence of non-singular standardBT for the equation (10), where ζ(z) = 0, e.g. for any equation of the form

z1 − f(z) · z22 − ξ(z) · (z2)2 − η(z) · z2 = 0 (f(z) 6= 0). (12)

We point out, in particular, the standard non-singular BT for the differential equa-tion (12) with the following Backlund system

y1 = (ay2 + by + c) ·(z2 · fξ(z)−

∫ η(z)·fξ(z)f(z) dz

),

y2 = (ay2 + by + c) ·(∫ fξ(z)

f(z) dz + 1) ,

where a, b, c = const (a2 + b2 + c2 6= 0). It is the standard BT for the differentialequation (12). In general, this BT is non-singular.

It is proved also the existence of standard Backlund transformations for theequation (10), where ζ(z) is of the form (11) and moreover A2 + B2 6= 0. Such BTis for example the BT with the following Backlund system

Y1 = BY + a ·(z2 · fξ(z)−

∫ η(z)·fξ(z)f(z) dz −A

∫fξ(z)dz + k1

),

Y2 = −AY + a ·(∫ fξ(z)

f(z) dz + k2

) ,

where a, k1, k2 = const and moreover a 6= 0 and k1, k2 satisfy the relation Ak1 +Bk2 − C = 0.

References

[1] Rybnikov A.K. Special Connections that determine the Representation of Zero Curvature for

Second-Order Evolution Equations, Izv. Vuz. Math. No.9, (1999), pp. 32-41 (in Russian).[2] Rybnikov A.K. The Theory of Connections and the Problem of the Existence of Backlund

Transformations for Second-Order Evolution Equations, Doklady Mathematics. Vol.71 No.1.,(2005), pp. 71-74.

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On the theory of smooth LM-loopsLiudmila Sabinina Soboleva

(Universidad Autonoma del Estado de Morelos Cuernauoea, Mexico)We would like to discuss the properties of smooth LM-loops. In particular the dif-ferential equation of such loops will be presented. Smooth left Bol loops, smooth leftCC-loops and LM-loops, which admit the transsymmetric structure are subclassesof the class of smooth LM-loop.

The Hodge spectrum and convexityAlessandro Savo (University of Rome, La Sapienza)

We discuss the first positive eigenvalue µ[p]1 of the Hodge Laplacian acting on smooth

p-differential forms of a Riemannian manifold with boundary, for the absoluteboundary condition. We give a geometric lower bound when the boundary hasa suitable degree of convexity. For convex euclidean domains, we show that the se-quence µ[p]

1 is not decreasing in the degree p, and we expose recent work on sharpgeometric bounds for such eigenvalues. Finally, we examine the inverse problem:knowing µ

[p]1 for all degrees p, what can be said about the shape of the domain?

Variational SequencesJana Sedenkova (Tomas Bata University in Zlin)

The variational sequences and the interior Euler-Lagrange operator are used togeneralize the concept of a Lepage form, introduced by Krupka, to field theory.These forms allow us to find a suitable representation of the classes of forms invariational sequences in field theory.

Imprimitive groups sharply 2-transitive on blocksKarl Strambach (Universitat Erlangen-Nurnberg, Mathematisches Institut )

Let G be an imprimitive locally compact group acting on a locally compact con-nected topological space such that the set S of blocks forms a locally compact space.I want to classify all groups G acting on the manifold S as well as on any blocksharply 2-transitively and satisfying the following two conditions:

(a) The inertia group N of G is 2-dimensional and induces a sharply transitivegroup on each block;

(b) if ∆1 , ∆2 are two distinct blocks and xi, yi are points in ∆i, i = 1, 2, thenthere is just one element in N mapping xi onto ji, for i = 1, 2.

The classical example for such groups is the group of affine mappings of the affineline over the ring of dual numbers.

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The nonholonomic constraint structure of higher orderMartin Swaczyna (Department of Mathematics, University of Ostrava)

We generalize a nonholonomic constraint structure of the first order on constraintssubmanifolds in higher order jet spaces. The annihilator of a canonical distributiongenerates a constraint ideal. We study differential forms on a constraint submani-fold of order r due to presence of the constraint ideal. Finner classification of theconstraint ideal is presented, the horizontalization and contactization mappings aregeneralized, and canonical decomposition of forms is found.

Isometric actions of compact Lie groups on globally hyperbolic Lorentzmanifolds

David Szeghy (Eotvos University, Budapest)Let a compact connected Lie group G act isometrically on a globally hyperbolicLorentz manifold L. The correspondance between the focal locus of an orbit withmaximal orbit type and the singular orbits will be examined. Moreover, it willbe shown that if there is an orbit of codimension 1 then every orbit is a Cauchyhypersurface which is diffeomorphic with G and L is diffeomorhic to a productspace G× (a, b).

Quasi linear connections in TM for Randers spacesL. Tamassy (Mathematics Institute of Debrecen University)

A Finsler space is a manifold endowed with a Finsler metric F(x, y), x ∈ M ,y ∈ TxM . F(x, y) = ‖y‖ is the Finsler norm of y ∈ TxM . In the simple case‖y‖ = ‖ − y‖. In the general case this is not true. In the first case the indicatrixI(x0) = y ∈ Tx0M | ‖y‖ = 1 is symmetric on x0 = O, in the general case it isnot.

In most Finsler spaces Fn = (M,F) there exists no metrical and linear connec-

tionm`

Γ (x) among the vectors y(x) | x ∈ TxM. We showed that such connectionsm`

Γ (x) exist exactly in those Finsler spaces Fn, which are affine deformations oflocally Minkowski spaces: Fn = A `Mn. The indicatrices I(x) of every Fn aresymmetric. Translating every I(x) ⊂ TxM in TxM from the origin O of TxM by avector c(x) we obtain a new indicatrix I(x), which is no more symmetric on O (itis symmetric on the endpoint C of c(x)). These spaces Fn = (M, F) do not have am`

Γ (x), but – as we show –

Γik(x, y, c) = Γj

ik(x)yj − ‖y‖

[Γj

ik(x)cj − ∂ci

∂xk

](1)

is a (simple) metrical connection among the vectors y(x) | x ∈ TxM. Here

Γjik(x) are the coefficients of the

m`

Γ (x) of that Fn from which the Fn originates. Γis homogeneous in y, the bracket is linear in c and independent of y. The curvaturetensor of (1) can also be calculated, and it yields a simple expression.

Randers spaces are the simplest case of the Fn, where the indicatrices are trans-lated ellipsoids. Thus Randers spaces originate from Fn which are Riemann spaces.Randers spaces have important applications in electron optics and in the theory ofthe electron microscope.

28

References

[1] Bao D. – Chern S.S., A note on the Gauss-Bonnet theorem for Finsler spaces, Ann of Math.143 (1996), 233–252.

[2] Bao D. – Robles C. – Shen Z., Zermelo navigation on Riemannian manifolds, J. Diff. Geom.

66 (2004), 377–435.[3] Hrimiuc D., On the affine deformation of a Minkowski norm, Per. Math. Hungar 48 (2004),

49–60.

[4] Tamassy L., Point Finsler spaces with metrical linear connections, Publ. Math. Debrecen 56(2000), 643–655.

Integral geometry in Riemannian symmetric spacesHiroyuki Tasaki (University of Tsukuba)

Integral formulae of integral geometry in real space forms like as Crofton andPoincare formulae and various kinds of kinematic formulae are generalized in thecase of Riemannian homogeneous spaces by R. Howard.

In this talk we show more explicit expressions of such integral formulae in Rie-mannian symmetric spaces by the use of some extrinsic geometric amounts of sub-manifolds which are invariant under the actions of the isotropy subgroups.

Variationally complete actionsGudlaugur Thorbergsson (Mathematisches Institut der Universitat zu Koln)I will review the basic properties of variationally complete and polar actions andthen explain how these concepts can be extended to singular Riemannian foliations.In the last part of the talk I will present a new result saying that a variationally com-plete action on a nonnegatively curved Riemannian manifold is hyperpolar (jointwork with Lytchak).

On metrizability of linear connections on smooth or analytic manifoldsO. Krupkova, A. Vanzurova, Z. Vilimova

(Dept. Algebra and Geometry, Palacky University)Our aim is to contribute to the problem of metrizability of a linear connectionconcerning the following questions:

(1) For a torsion-free linear connection∇ on a smooth manifold M , find necessaryand sufficient conditions for existence of a metric g on M such that ∇ is just theLevi-Civita connection of (M,∇).

(2) In the positive case, describe all metrics belonging to a given torsion-freelinear connection (g need not be unique).

(3) In the positive, try to express the components gij of g by means of Christoffels,or components of the curvature tensor, by an explicit formula.

While (1) was answered already in 1922 by L.P. Eisenhart and O. Veblen, anddiscussed later on by many authors, e.g. O. Kowalski, B.G. Schmidt, M. Anastasiei,(3) seems to be still open.

We give a survey of various methods used in this respect, and demonstrateadvantages of algorithms developed for this purpose on simple examples.

29

On F-planar mappings with certain initial conditionsMikes Josef (Palacky University Olomouc),

Pokorna Olga (Czech University of Agriculture),Vavrıkova-Chuda Hana (Tomas Bata University in Zlin)

In this paper we investigate F2-planar mappings of n-dimensional (pseudo-) Rie-mannian spaces Vn → Vn with metric satisfying at a finite number of points follow-ing conditions g(X, Y ) = f g(X, Y ). It comes out that even under this conditionsit holds that F2-planar mapping is homothetic.

Osserman metrics on Walker 4-manifolds equipped with apara-Hermitian structure

J. Carlos Dıaz-Ramos, Eduardo Garcıa-Rıo, Ramon Vazquez-Lorenzo(University of Santiago de Compostela)

The study of the curvature is a central topic in Riemannian and pseudo-Riemanniangeometry, as it provides of the simplest algebraic invariant of the metric structure.Since the whole curvature tensor is difficult to handle, a typical technique consistsin dealing with natural operators associated to the curvature, the Jacobi operatorbeing the most natural and widely investigated. A pseudo-Riemannian manifold(M, g) is said to be Osserman if the eigenvalues of the Jacobi operators are constanton the unit pseudo-sphere bundles. Any two-point homogeneous space is Osserman,and the converse is also true in the Riemannian (dim M 6= 16) and Lorentziansettings. However, many examples of nonsymmetric Osserman pseudo-Riemannianmanifolds are known in other signatures.

All known examples of Osserman manifolds have either diagonalizable or nilpo-tent Jacobi operators, which was conjectured to be true in the general case byseveral authors. The purpose of this talk is to report on a new family of Ossermanmanifolds whose Jacobi operators are neither diagonalizable nor nilpotent, thusshowing that the structure of Osserman manifolds in indefinite signature is sub-tler than expected. More precisely, we investigate a natural almost para-Hermitianstructure defined on a Walker four-dimensional manifold, and the Einstein equationslead to three families of metrics: one of them does not give Osserman manifolds,the second one consists of Osserman manifolds with Jacobi operators either van-ishing or nilpotent, and finally, the third family of Einstein para-Hermitian Walkermanifolds provides examples of Osserman four-dimensional manifolds with exactlytwo distinct eigenvalues of the Jacobi operators, α and β = 4α 6= 0, the formerwith multiplicity two and, therefore, it is a double root of the minimal polynomialof the Jacobi operators.

Cohomogeneity one isometric actions on exceptional compactsymmetric spaces

Laszlo Verhoczki (Eotvos University, Budapest)We discuss in detail cohomogeneity one isometric actions on the exceptional com-pact symmetric spaces G2, G2/SO(4), F4 and F4/(Sp(3) × Sp(1)). This talk isconcerned with the isometric actions of those compact Lie groups where the prin-cipal orbits coincide with the tubular hypersurfaces around the totally geodesicsingular orbit. This means that the symmetric spaces can be thought of as com-pact tubes. We determine the radii of the tubes and the shape operators of the

30

principal orbits. In the case of G2 and G2/SO(4), it turns out that the shape oper-ators and the Jacobi operators acting on the tangent spaces of the principal orbitsdo not commute, that is, the orbits of codimension one are not curvature-adaptedsubmanifolds. Finally, the volumes of the principal orbits and the volumes of thesymmetric spaces will be computed in terms of the maximal sectional curvature.

On an existence theorem of Wagner manifoldsCsaba Vincze (University of Debrecen)

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997),307-321] S. Bacso, M. Hashiguchi and M. Matsumoto give a condition for a 1-formβ to be the perturbation of a Riemannian manifold (M,α) such that the manifoldequipped with any (α, β)-metric is a Wagner manifold with respect to the Wagnerconnection induced by β. The condition shows that its covariant derivative withrespect to the Levi-Civita connection must be of a special form

(∇β)(X, Y ) = ‖β]‖2α(X, Y )− β(X)β(Y ).

We are going to give a family of Riemannian manifolds admitting nontrivial solu-tions. It will be also proved that there are no further essentially different examples;in particular we consider the classical hyperbolic space together with a nontrivialsolution.

Non-holonomic systemsPetr Volny (VSB - Technical University of Ostrava)

A generalization of the concept of a system of non-holonomic constraints to fiberedmanifolds with n-dimensional bases is considered. Motion equations in both La-grangian and Hamiltonian setting for systems subjected to such constraints areinvestigated. Regularity conditions for the existence of a non-holonomic Legendretransformation, and the corresponding formulas for Hamiltonian and momenta arefound. In particular, semiholonomic constraints and simplifications arising in thiscase are discussed.

Integral formulae for foliations and non-integrable distributionsPawel Walczak (Uniwersytet Lodzki)

We shall provide a survey of a variety of integral formulae which appeared in thefoliation theory during last 50 years: from Reeb’s∫

M

h = 0

with h being the mean curvature of a codimension-one foliation of a compact Rie-mannian manifold M to the speaker’s∫

M

(K(D1, D2) + ‖A1‖2 − ‖H1‖2 − ‖T1‖2

+ ‖A2‖2 − ‖H2‖2 − ‖T2‖2)

= 0,

where D1 and D2 are complementary orthogonal distributions on a compact Rie-mannian manifold M , Ai, Hi and Ti are, respectively, the second fundamentalform, the mean curvature vector and the integrability tensor of Di (i = 1, 2), whileK(D1,K2) is so called mixed curvature in the direction of D1 and D2. We shall

31

discuss several recent applications of the formulae: (1) to the problem of prescribingmean curvatures for foliations (due to Oshikiri, 1997),(2) to energy of unit vectorfields (due to Brito and the speaker, 2000),(3) to the conformal geometry of folia-tions of negatively curved manifolds (due to Langevin and the speaker, 2004), (4)to holomorphic foliations (due to Svensson, 2003).

Hamiltonian structure of gauge-natural field theoriesM. Palese and E. Winterroth

(University of Torino)We consider the second variational derivative of a given gauge-natural invariantLagrangian taken with respect to (prolongations of) vertical parts of gauge-naturallifts of infinitesimal principal automorphisms. By requiring such a second varia-tional derivative to vanish, via the Second Noether Theorem we find that a covariantstrongly conserved current is canonically associated with the deformed Lagrangianobtained by contracting Euler–Lagrange equations of the original Lagrangian with(prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal au-tomorphisms lying in the kernel of the generalized gauge-natural Jacobi morphism.This conserved current can be suitably interpreted as an Hamiltonian associatedwith the deformed Lagrangian.

Complex Flag Manifolds and Representations of Semisimple Lie GroupsJoseph A. Wolf

(University of California at Berkeley)It has been known for about half a century that representations of compact Liegroups can be constructed in the beautiful complex–geometric settting of the Borel–Weil theorem and (more cohomologically) the Bott–Borel–Weil Theorem. About 40years ago, geometric quantization was developed by Kostant, Kirillov and Souriau,establishing the link between physics and geometric representation theory. Thequestion then: what about noncompact Lie groups? For discrete series represen-tations this was essentially settled by Schmid in the early 1970’s, and then in themid 1970’s I worked out the solution for the other tempered series of representa-tions of real reductive Lie groups. After that, the question became: what aboutnon–tempered representations?

I’ll sketch the geometric setting for these realizations of representations of realreductive Lie groups and indicate how the cycle space and the double fibrationtransforms are used for representations that need not be tempered. The doublefibration transforms considered here, carry cohomology of holomorphic vector bun-dles to spaces of holomorphic functions, in a manner equivariant for the action ofa reductive Lie group. The best known example is the complex Penrose transform.In the last couple of years there has been a lot of progress on the general theoryfor the double fibration transform from holomorphic vector bundles on a flag do-main. This talk is an indication of both the background and the current state ofthe theory.

32

Jacobi fields along harmonic mapsJohn C Wood (University of Leeds, UK)

A harmonic map between Riemannian manifolds is a map which extremizes a cer-tain natural energy functional; they appear in particle physics as nonlinear sigmamodels. A Jacobi field is an infinitesimal deformation of a harmonic map. It isimportant to know whether the Jacobi fields along the harmonic maps betweengiven Riemannian manifolds are integrable, i.e., arise from variations through har-monic maps. If they do, then the space of harmonic maps is a smooth manifoldwith tangent space given by the Jacobi fields; we also gain some information onthe structure of the singular set of weakly harmonic maps. We shall outline whatis known about Jacobi fields and their integrability starting with closed geodesicsand moving on to harmonic maps from surfaces.

Laplacian for Riemannian Curvature TensorMakoto Yawata and Hideko Hashiguchi

(Department of Mathematics, Chiba Institute of Technology)Let (M ; g) be a Riemannian manifold with metric tensor g and R its Riemanniancurvature tensor. Here we define the tensor fields H of type (1, 5) by

H(X, Y, Z,W ) = (R(X, Y )R)(Z,W ) + (R(Z,W )R)(X, Y ).

Then H satisfies the following conditions;

H(X, Y, Z,W ) = −H(Y, X,Z,W ),

H(X, Y, Z,W ) = −H(X, Y,W,Z),

H(X, Y, Z,W ) + H(Y, Z, X,W ) + H(Z,X, Y,W ) = 0,

H(X, Y, Z,W,U, V ) + H(Z,W,U, V,X, Y ) + H(U, V,X, Y, Z, W ) = 0.

H is a proper generalized curvature tensor of type (1, 5). If we put Hhijkuv =g(H(∂h, ∂i, ∂j , ∂k)∂u, ∂v) then the local components of tensor H are given by

Hhijkuv =−RhijaRakuv −Rhik

aRjauv

−RjkhaRaiuv −Rjki

aRhauv

The component Hbhijka is given by

Hbhjkib = Bhijk −RaiRjkh

a.

Here we have

Bhijk = −RhiabRbjk

a −RbjhaRaik

b + RbjiaRahk

b.

Next, we define a linear map τ from L15 to R1

3, τ(H) is given by

Hhijk = 4Bhijk −RhaRaijk −Ri

aRhajk −RjaRhiak −Rk

aRhija.

We can also see that the Ricci curvature tensor Ric(H) of H is given by

Hij = 2(RabRaijb −RiaRaj),

and the scalar curvature of H vanishes.We also define the tensor S = (Shijk) by

Shijk = ∇hSjki −∇iSjkh +∇jShik −∇kShij ,

33

where Shij = ∇hRij −∇iRhj .Theorem. Let (M, g) be a Riemannian manifold. Then we have the following

formula:2∆Rhijk + Shijk + Hhijk = 0.

Corollary. Let (M, g) be a Riemannian manifold. Then Shij satisfies the fol-lowing formula:

∇jShik +∇hSijk +∇iSjhk = RhikaRja + Rijk

aRha + RjhkaRia.