Submanifolds in Symmetric Spaces

Jurgen BerndtUniversity College Cork

OCAMI-KNU Joint Workshop on Differential Geometry and Related Fields”Submanifold Geometry and Lie Theoretic Methods”Osaka City University, 30 October - 3 November 2008

Overview

I. Euclidean spaces

1. Hypersurfaces with constant principal curvatures2. s-representations3. Normal holonomy4. Polar representations

II. Symmetric spaces

1. Cohomogeneity one actions: general theory2. Cohomogeneity one actions: classifications3. Hypersurfaces with constant principal curvatures4. Hyperpolar and polar actions

General problems in submanifold geometry

I Find “good” geometric invariants for submanifolds , e.g.I principal curvaturesI mean curvature

I Classify submanifolds according to these invariants , e.g.I all principal curvatures vanish: classification of totally geodesic

submanifoldsI all principal curvatures are constant: classification of

isoparametric submanifoldsI mean curvature vanishes: classification of minimal

submanifolds

We discuss a particular example:

Surfaces in E3

Problem. What are the surfaces in E3 having the same shapeeverywhere? In other words, what are the surfaces in E3 withconstant principal curvatures?

I Examples: affine planes, round cylinders, round spheres

I Question: Are there others?

I Let’s investigate this problem

I Let P ⊂ E3 be a surface with constant principal curvaturesλ1, λ2 and unit normal field ξ

Surfaces in E3 (II)

Case 1: λ1 = λ2 =: λ (=⇒ P umbilical surface in E3)

I λ = 0 =⇒ P totally geodesic =⇒ P open part of an affineplane in E3

I λ > 0. Consider

F : P → E3, p 7→ p +1

λξp

Then

DpF (u) = u +1

λ(−Aξp u) = 0

for all u ∈ TpP; A shape operator of P. Thus F is constant,say F (p) = z ∈ E3. Conclusion: P lies on the sphere withcenter z and radius 1/λ

Surfaces in E3 (III)

Case 2: λ1 6= λ2

I Choose local unit vector fields E1,E2 ∈ Γ(TP) withAEi = λiEi

I Then the Codazzi equation implies

(λi − λj)〈∇PEi

Ei ,Ej〉 = 〈(∇PEi

A)Ei ,Ej〉= 〈(∇P

EjA)Ei ,Ei 〉 = 0

and hence∇P

EiEi = 0

I The Codazzi equation also implies

λ1∇PE2

E1 − A∇PE2

E1 = (∇PE2

A)E1 = (∇PE1

A)E2

= λ2∇PE1

E2 − A∇PE1

E2

and hence

A[E1,E2] = λ2∇PE1

E2 − λ1∇PE2

E1

Surfaces in E3 (IV)

I Now use the Gauß-Codazzi equations for

λ1λ2 = 〈RP(E1,E2)E2,E1〉= 〈∇P

E1∇P

E2E2,E1〉 − 〈∇P

E2∇P

E1E2,E1〉 − 〈∇P

[E1,E2]E2,E1〉

= −E2〈∇PE1

E2,E1〉+ 〈∇PE1

E2,∇PE2

E1〉

− 1

λ2 − λ1〈(∇P

[E1,E2]A)E2,E1〉

=1

λ1 − λ2〈(∇P

E1A)E2, [E1,E2]〉

=1

λ1 − λ2(λ2〈∇P

E1E2, [E1,E2]〉 − 〈∇P

E1E2,A[E1,E2]〉)

= 0

I Thus λ1 = 0 or λ2 = 0. Assume λ2 = 0 and λ1 > 0.

Surfaces in E3 (V)

I Define

F : P → E3, p 7→ p +1

λ1ξp

I Then

DF (E1) = 0 , DF (E2) =

(1− λ2

λ1

)E2 = E2

I Thus F has constant rank one and hence for each point p ∈ Pthere exists an open neighborhood U of p in P such thatF : U → E3 is a submersion onto a one-dimensionalsubmanifold Q

I P lies on the tube of radius 1/λ1 around Q

What is the geometry of Q?

Surfaces in E3 (VI)

I Let c be an integral curve of E2

I Then

α(c , c) = 〈α(c, c), ξ ◦ c〉ξ ◦ c = 〈Aξ c , c〉 = 0

and∇P

c c = (∇PE2

E2) ◦ c = 0

I Thus c is a geodesic in E3 by Gauß formula, hence a straightline segment

I Moreover,∇E3

c ξ = −Aξ c = 0,

which shows that ξ is parallel along c

I Thus F ◦ c is a straight line segment as well

I Altogether this implies that Q is a straight line segment, andhence P lies on a round cylinder in E3

Thus we have proved the following result:

Surfaces in E3 (VII)

Somigliana 1919: A connected surface in E3 has constantprincipal curvatures if and only if it is an open part of

I a plane in E3, or

I a sphere of radius r ∈ R+ in E3, or

I a tube of radius r ∈ R+ around a line E3

It is remarkable that the surfaces listed above are precisely thehomogeneous surfaces in E3. WHY?

Hypersurfaces in En

Segre 1938: A connected hypersurface in En has constantprincipal curvatures if and only if it is an open part of ahomogeneous hypersurface:

I an affine hyperplane in En, or

I a tube around an affine subspace of En of dimensionk ∈ {0, . . . , n − 2}

Again, it is remarkable that the hypersurfaces listed above areprecisely the homogeneous hypersurfaces in En. WHY?

Main ingredients of proof

I Fundamental equations of submanifold theory

I Focal set theory

Fundamental equations of submanifold geometry

P submanifold of Riemannian manifold M

I Gauss formula: ∇MX Y = ∇P

X Y + α(X ,Y )

I Weingarten formula: ∇MX ξ = −AξX +∇⊥X ξ

I Gauss equation:〈RP(X ,Y )Z ,W 〉 =〈RM(X ,Y )Z ,W 〉+〈α(Y ,Z ), α(X ,W )〉−〈α(X ,Z ), α(Y ,W )〉

I Codazzi equation:

(RM(X ,Y )Z )⊥ = (∇⊥Xα)(Y ,Z )− (∇⊥Yα)(X ,Z )

I Ricci equation:

〈RM(X ,Y )ξ, η〉 = 〈R⊥(X ,Y )ξ, η〉 − 〈[Aξ,Aη]X ,Y 〉

Focal set theory

The geometry of focal sets in Riemannian geometry can be studiedusing Jacobi fields

I P hypersurface of Riemannian manifold M

I ∀p ∈ P : TpM = TpP ⊕ νpP

I ξ (local) unit normal vector field of P

I γ geodesic in M with p = γ(0) ∈ P and ξp = γ(0)

I RMγ = RM(·, γ)γ Jacobi operator along γ

I A vector field Y along γ is called a Jacobi field if

Y ′′ = RMγ ◦ Y

I A Jacobi field Y along γ is called a P-Jacobi field if

Y (0) ∈ TpP and Y ′(0) + AξY (0) ∈ νpP

I Y0(t) = tγ(t) is a P-Jacobi field along γ

Focal set theory (II)

I J(P, γ) = real vector space of all P-Jacobi fields along γwhich are perpendicular to Y0

I dim J(P, γ) = dim P

I expP : νP → M normal exponential map of P

I Pr = {expP(rξp) | p ∈ P} displacement of P in direction ξ atdistance r ∈ R+

I Assume that Pr is a submanifold of M

I Ar shape operator of Pr with respect to γ(r)

Theorem.

Tγ(r)Pr = {Y (r) | Y ∈ J(P, γ)} and Ar Y (r) = −(Y ′(r))T

Symmetric spaces

A connected Riemannian manifold M is called a symmetric spaceif for all p ∈ M there exists an isometry sp of M such that

I s2p = idM

I p is an isolated fixed point of sp

Examples:

I M = En: sp = reflection in p

I M = Sn ⊂ En+1: sp = reflection in line through p and 0

Basics about symmetric spaces

Let M be a symmetric space

I M is homogeneous, i.e., the group G = I (M) of isometries ofM acts transitively on M

I The Riemannian curvature tensor of M is parallel

I The Riemannian universal covering space M is a symmetricspace with deRham decomposition

M = M0 × M1 × · · · × Mk ,

whereI M0 = En0 for some n0 ∈ Z≥0; M0 is called the Euclidean factor

of MI Mk , k ≥ 1, is a simply connected irreducible symmetric space

If n0 = 0 then M is called a semisimple symmetric spaceIf (n0, k) = (0, 1) then M is called a simple symmetric space

Basics about symmetric spaces (II)

I If M is semisimple, then the Lie algebra g of G = I (M) issemisimple, i.e., the Killing form

B : g× g→ R , (X ,Y ) 7→ tr(ad(X ) ◦ ad(Y ))

is non-degenerate

I If M is simple, then the Lie algebra g of G = I (M) is simple,i.e., g is semisimple and the only ideals in g are {0} and g , org is the direct sum of two simple Lie algebras

I If M is simple then the sectional curvature of M is either ≥ 0or ≤ 0

I If M is simple and the sectional curvature of M is ≥ 0, thenM is compact with finite fundamental group

I If M is simple and the sectional curvature of M is ≤ 0, thenM is noncompact and simply connected

Basics about symmetric spaces (III)

I A symmetric space is said to be of compact type if it issemisimple and the sectional curvature is ≥ 0

I A symmetric space is said to be of noncompact type if it issemisimple and the sectional curvature is ≤ 0

I Symmetric spaces have been classified by Elie Cartan usingthe classification of simple Lie algebras, their involutiveautomorphisms, and the concept of duality betweensymmetric spaces of compact type and of noncompact type

Basics about symmetric spaces (IV)

Let M be a semisimple symmetric space, o ∈ M, G = I o(M),K = {g ∈ G | g(o) = o}. Then M can be identified with thehomogeneous space G/K .

I p = {X ∈ g | B(X , k) = {0}}I g = k⊕ p Cartan decomposition

I θ : g→ g,X + Y 7→ X − Y Cartan involution (X ∈ k andY ∈ p)

I [k, k] ⊂ k , [k, p] ⊂ p , [p, p] ⊂ k

The rank of M is the dimension of a maximal abelian subspace ofp. Symmetric spaces of rank one:

I The spheres Sn, n ≥ 2

I The projective spaces RPn, CPn, HPn, OP2, n ≥ 2

I The hyperbolic spaces RHn, CHn, HHn, OH2, n ≥ 2

s-representations

An s-representation is the isotropy representation of a simplyconnected semisimple symmetric space M = G/K

χ : K × p→ p, (k ,X ) 7→ Ad(k)X

An orbit K · X = {Ad(k)X | k ∈ K} of an s-representation iscalled a real flag manifold or R-space.

The orbits of s-representations provide an important class ofsubmanifolds in Euclidean spaces, which takes us to the concept ofnormal holonomy...

Holonomy of Riemannian manifolds

I Mn connected Riemannian manifold

I Hol(M) holonomy group of M

I Hol0(M) restricted holonomy group of M

General facts:

I Hol0(M) connected compact subgroup of SOn

I Hol0(M) trivial ⇐⇒ M flat

I Hol(M) reducible ⇐⇒ M is Riemannian product

I Berger 1953, Simons 1962: If Hol(M) is irreducible anddoes not act transitively on Sn−1, then M is locally symmetric(∇R = 0)

Normal holonomy of submanifolds

I Mn connected Riemannian manifold

I Pk connected submanifold of M

I Recall Weingarten formula

∀X ∈ Γ(TP), ξ ∈ Γ(νP) : ∇MX ξ = −AξX +∇⊥X ξ

I ∇⊥ normal connection of P

I ∇⊥-parallel translation of normal vectors ξ ∈ νpP for fixedp ∈ P along piecewise smooth curves c : [0, 1]→ P withc(0) = c(1) = p induces a subgroup Φp of O(νpP) ∼= On−k

I Φp normal holonomy group of P at p

I p, q ∈ M =⇒ Φp and Φq are conjugate

Normal holonomy of submanifolds (II)

I ∇⊥-parallel translation of normal vectors ξ ∈ νpP for fixedp ∈ P along null homotopic piecewise smooth curvesc : [0, 1]→ P with c(0) = c(1) = p induces a Lie subgroupΦ∗p of SO(νpP) ∼= SOn−k

I Φ∗p restricted normal holonomy group of P at p

I Φ∗p is the identity component of Φp

I Φ∗p is a normal subgroup of Φp and Φp/Φ∗p is countable

Normal holonomy of surfaces in E3

Let P2 be a surface in E3. Then

I Φp = {I} ⇐⇒ P2 orientable

I Φp = Z2 ⇐⇒ P2 non-orientable

Normal holonomy of curves in E3

I c : [0, l ]→ E3 smooth curve with c(0) = c(l) and ||c || = 1

I κ curvature and τ torsion of c

I (T ,N,B) Frenet frame field along c

I N ′ = −κT − τB and B ′ = τN

I ξ unit normal vector field along c

I Write ξ = cos(ω)N + sin(ω)B

I Differentiate

ξ′ = −ω′ sin(ω)N + cos(ω)N ′ + ω′ cos(ω)B + sin(ω)B ′

= −κ cos(ω)T + (τ − ω′) sin(ω)N − (τ − ω′) cos(ω)B

I Thus

∇⊥ξ = 0⇐⇒ ω′ = τ ⇐⇒ ω(t) =

∫ t

0τ(s)ds + C

Normal holonomy of curves in E3 (II)

I Thus

∇⊥ξ = 0⇐⇒ ω′ = τ ⇐⇒ ω(t) =

∫ t

0τ(s)ds + C

I Define

ω =

∫ l

0τ(s)ds = total torsion of c

I Conclusion:Φp∼= {e ikω | k ∈ Z}

Remark: ∀r ∈ R ∃c : [0, l ]→ E3 :∫ l

0 τ(s)ds = rTherefore the possible normal holonomy groups of closed curves inE3 are

Z and Zk (k ∈ Z>0)

Normal Holonomy Theorem

Olmos 1990: Let P be a submanifold of a space form (En, Sn,RHn), p ∈ P and Φ∗ = Φ∗p. Then

I Φ∗ is compact

I There exist a unique (up to order) orthogonal decomposition

νpP = V0 ⊕ V1 ⊕ · · · ⊕ Vk

of νpP into Φ∗-invariant subspaces V0, . . . ,Vk and normalsubgroups Φ0, . . . ,Φk of Φ∗ such that

I Φ∗ = Φ0 × Φ1 × · · · × Φk (direct product)I Φi acts trivially on Φj (i 6= j)I Φ0 = {I} and Φi (i ≥ 1) acts on Vi as an irreducible

s-representation

The proof is based on Ambrose-Singer Holonomy Theoremand Holonomy Systems

Ambrose-Singer Holonomy Theorem

The Ambrose-Singer Holonomy Theorem relates holonomywith curvature

For c : [0, 1]→ P piecewise smooth curve define

I τc parallel transport along c in P

I τ⊥c ∇⊥-parallel transport along c

Ambrose-Singer 1953: The Lie algebra of Φ∗ is the subalgebra ofso(νpP) generated by all transformations of the form

(τ⊥c )−1 ◦ R⊥τcX ,τcY ◦ τ⊥c

where c : [0, 1]→ P is a piecewise smooth curve with c(0) = pand X ,Y ∈ TpP

Holonomy systems

Holonomy systems were introduced by Simons 1962

A holonomy system [V ,R,G ] consists of

I V n-dimensional Euclidean vector space

I R algebraic curvature tensor on V

I G connected compact subgroup of SO(V ) such that

∀x , y ∈ V : Rx ,y ∈ g

Example: If (M, g) is a Riemannian manifold and p ∈ M, then[TpM,Rp,Hol0(M, p)] is a holonomy system

The idea for proof of Normal Holonomy Theorem

I Use arguments given by Simons 1962 for proof of BergerHolonomy Theorem

I Problem: Find suitable holonomy system [νpP, ?,Φ∗p]

I Note that R⊥p : ∧2TpP → ∧2νpP

I Define R⊥p = R⊥p ◦ (R⊥p )∗ : ∧2νpP → ∧2νpP

I Ricci equation: 〈R⊥x ,yξ, η〉 = 〈[Aξ,Aη]x , y〉

〈R⊥p (ξ1, ξ2)ξ3, ξ4〉 = 〈(R⊥p )∗(ξ1 ∧ ξ2), (R⊥p )∗(ξ3 ∧ ξ4)〉= −tr([[Aξ1 ,Aξ2 ], [Aξ3 ,Aξ4 ]])

R⊥p is an algebraic curvature tensor on νpP (with negative scalar

curvature) and [νpP,R⊥p ,Φ∗p] is a holonomy system

Some problems

I The definition of the holonomy system uses the Ricci equationand works only for space forms. How can one define aholonomy system for other Riemannian manifolds M?

I Which irreducible s-representations can be realized as therestricted normal holonomy representation of a submanifold ofa Euclidean space?Heintze-Olmos 1992: Investigated normal holonomy oforbits of s-representations: All s-representations arise asnormal holonomy representations with 11 exceptions. Nodefinite answer yet for any of the 11 exceptions

Complex submanifolds in complex projective spaces

Console-DiScala-Olmos 2008: Let P be a full complete complexsubmanifold of CPn. Then the normal holonomy group does notact transitively on the unit sphere in the normal space if and only ifP is the projectivized (unique) complex orbit of thes-representation of an irreducible Hermitian symmetric space ofrank ≥ 3.

Remark: Any such complex submanifold of CPn has parallel secondfundamental form; classification by Nakagawa-Takagi 1976

Geometry of orbits of s-representations

I Every singular orbit of an s-representation is a submanifoldwith constant principal curvatures: The principalcurvatures are constant for any parallel normal vector fieldalong any piecewise smooth curve

I Every principal orbit of an s-representation is anisoparametric submanifold: submanifold with constantprincipal curvatures and flat normal bundleThorbergsson 1991: The converse holds if P is connectedcomplete irreducible and codimP ≥ 3

I Heintze-Olmos-Thorbergsson 1991: A submanifold of En

has constant principal curvatures if and only if it isisoparametric or a focal manifold of an isoparametricsubmanifold

Polar representations on Euclidean spaces

A polar representation on En consists of

I a compact Lie group G acting isometrically on En

I an affine subspace Σ of En (called a section) with theproperties

I ∀x ∈ En : (G · x) ∩ Σ 6= ∅I ∀x ∈ Σ : Σ ∼= TxΣ ⊂ νx(G · x)

Examples:

I SO2 on E2

I Every s-representation is polar and every maximal abeliansubspace a of p is a section.

Dadok 1985: Every polar representation on En is orbit equivalentto an s-representation

Summary

On Euclidean spaces there is a beautiful theory of submanifoldsinvolving

I Symmetric spaces

I Polar representations

I Submanifolds with constant principal curvatures

Plan for what follows: Discuss some generalizations of this theoryfrom Euclidean spaces to symmetric spaces

Polar actions and hyperpolar actions

Let M be a connected complete Riemannian manifold and H be aconnected closed subgroup of I (M)

The action of H on M is said to be polar if there exists aconnected complete submanifold S of M such that

I ∀p ∈ M : (H · p) ∩ S 6= ∅I ∀p ∈ S : TpS ⊂ νp(H · p)

Any such submanifold S is called a section of the action

A polar action on M is called hyperpolar if there exists a flatsection.

Every section of a polar action is totally geodesic

Consequence: Every polar action on En is hyperpolar

Basic examples of hyperpolar actions

I Every s-representation induces a hyperpolar action on aEuclidean space

I Let k ∈ {1, . . . , n − 1} and consider H = Ek acting on En bytranslations. The action of H on En is hyperpolar andS = En−k = (Ek)⊥ is a section. The orbits form a foliation onEn by parallel affine subspaces

I Let M = G/K be a semisimple symmetric space. Then theaction of K on M is hyperpolar

I Every cohomogeneity one action is hyperpolar

Fundamental problem: Classification of polar actions and ofhyperpolar actions on symmetric spaces

Transformation groups

Motivation:

I Study of symmetries of mathematical or physical structures

Origin:

I Galois theory

I Felix Klein 1872: Geometry is the study of properties whichare invariant under a given transformation group

Transitive Actions

I Transitive action ⇐⇒ orbit space is a point

I Geometry of homogeneous spaces

I Fundamental property: Homogeneous spaces look the sameeverywhere

I Motivation: use algebraic methods for solving geometricproblems

Cohomogeneity One Actions

I Cohomogeneity one action (C1)⇐⇒ orbit space is one-dimensional

I Geometry of cohomogeneity one spaces

I Motivation: use ODE methods for solving geometric problems

I Submanifold geometry: Every homogeneous hypersurface isan orbit of a cohomogeneity one action

Examples of cohomogeneity one actions

The orbit space of a cohomogeneity one action

I M connected complete Riemannian manifoldI (M) isometry group of M

I H ⊂ I (M) connected closed subgroup acting on M withcohomogeneity one

I M/H = {H · p | p ∈ M} orbit space

Mostert 1957 (compact case)Berard Bergery 1982 (general case):

M/H ∼= R , S1 , [0,∞) or [0, 1]

I M/H ∼= R, S1

⇒ orbits form a Riemannian foliation on M

I M/H ∼= [0,∞), [0, 1]boundary point ←→ singular orbitinterior point ←→ principal orbit

Finite fundamental group

π1(M) finite ⇒ M/H 6∼= S1

Proof: use exact homotopy sequence for the fibre bundleF → M → M/H ∼= S1:

. . .→ π1(M)→ π1(S1) ≈ Z→ π0(F ) ≈ 0→ . . .

π1(M) = 0 and M compact ⇒ M/H ∼= [0, 1]

Hadamard manifolds

A connected, simply connected, complete Riemannian manifoldwith nonpositive sectional curvature is called a Hadamardmanifold

M Hadamard ⇒ M/H ∼= R or [0,∞)

Proof: Assume there exists a singular orbitLet K ⊂ H maximal compactCartan’s Fixed Point Theorem: ∃o ∈ M : K · o = o⇒ isotropy Ho = K and H · o singular orbit.Assume there exists second singular orbit H · p⇒ ∃h ∈ H : Hp ⊂ hKh−1 = hHoh−1 = Hh(o)

⇒ Hp fixes p and h(o)⇒ Hp fixes pointwise the geodesic in M from p to h(o)(Contradiction!)

Cohomogeneity one method

I M connected compact smooth manifold

I G compact Lie group acting on M with cohomogeneity one

I Assume M/G ∼= [0, 1], M0,M1 singular orbits

I M = M \ (M0 ∪M1) = union of principal orbits

Mdiff= (0, 1)× P with P a principal orbit

I g = dt2 + gt on Mgt one-parameter family of G -invariant metrics on P

Applications

I metrics of holonomy G2 or Spin7 (Gibbons, Pope, ...)

I hyperkahler and quaternionic Kahler structures (Dancer,Swann, ...)

I Einstein, Einstein-Kahler and Einstein-Weyl structures(Berard-Bergery, Bonneau, Dancer, Wang...)

I metrics with positive or nonnegative Ricci or sectionalcurvature (Grove, Wilking, Ziller, ...)

I harmonic maps, Yang-Mills equations (Urakawa, ...)

I special Lagrangian submanifolds (Joyce, Min-Oo, ...)

I differential topology (Atiyah-Berndt, ...)

Cohomogeneity one actions on spheres

Hsiang-Lawson 1971: Every cohomogeneity one action on asphere is orbit equivalent to the isotropy representation of asemisimple Riemannian symmetric space of rank 2

compact simple Riemannian symmetric spaces of rank 2:

SU3/SO3 , SU3 , SU6/Sp3 , E6/F4

Sp2 , SO10/U5 , E6/Spin10U1

SOn+2/SOnSO2 , SUn+2/S(UnU2) , Spn+2/SpnSp2

G2/SO4 , G2

Known classifications

Compact spaces:

I Takagi 1973: CPn

I Uchida 1977: H∗(M,Q) = H∗(CPn,Q)

I Iwata 1978: H∗(M,Q) = H∗(HPn,Q)

I Iwata 1981: H∗(M,Q) = H∗(OP2,Q)

I Kollross 2001: simply connected irreducible Riemanniansymmetric spaces of compact type and rank ≥ 2

Noncompact spaces:

I Segre 1938: En

I Cartan 1938: RHn

Euclidean spaces

I o(En) = En o SOn

Segre 1938: Every cohomogeneity one action on En is orbitequivalent to one of the following:

I H = En−1: orbits give a foliation by parallel hyperplanes

I H = Ek o SOn−k , 0 ≤ k < n− 1: orbits are a totally geodesicEk and the tubes around it

Proof: use Gauss-Codazzi equations and focal set theory

Real hyperbolic spaces

I o(RHn) = SOo1,n

Cartan 1938: Every cohomogeneity one action on RHn is orbitequivalent to one of the following:

I H = SOo1,n−1: orbits give a foliation by a totally geodesic

hyperplane RHn−1 and its equidistant hypersurfaces

I H = SOo1,k × SOn−k , 0 ≤ k < n − 1: orbits are a totally

geodesic RHk and the tubes around it

I H = N ⊂ KAN = SOo1,n: orbits give a foliation by horospheres

Proof: use Gauss-Codazzi equations and focal set theory

Restricted root space decomposition

I M = G/K , G = I o(M), Riemannian symmetric space ofnoncompact type

I B Killing form of g,B(X ,Y ) = tr(ad(X ) ◦ ad(Y))

I p = {X ∈ g | B(X , k) = 0}I g = k⊕ p Cartan decomposition

I θ : g→ g , X + Y 7→ X − Y Cartan involution

I 〈X ,Y 〉 = −B(X , θY ) positive definite inner product on g,induces Riemannian metric on M

I ad(X ) selfadjoint for all X ∈ p

I a ⊂ p maximal abelian subspace

I a∗ dual vector space of a

Restricted root space decomposition (II)

I gλ = {X ∈ g | ∀H ∈ a : ad(H)X = λ(H)X}, λ ∈ a∗

I 0 6= λ ∈ a∗ restricted root ⇐⇒ gλ 6= {0}I Σ ⊂ a∗ set of restricted roots

I a abelian ⇒ restricted root space decomposition

g = g0 ⊕⊕λ∈Σ

gλ , g0 = zk(a)⊕ a

I Σ ∈ {Ar ,Br ,Cr ,Dr ,E6,E7,E8,F4,G2,BCr} for M simple

I Λ = {α1, . . . , αr} set of simple roots of Σ

I Σ = Σ+ ∪ Σ−

Construction of Dynkin diagram

I Dynkin diagram: graph consisting of circles and lines, arrows

I circles represent simple roots

I angle between two simple roots is

π

2,

2π

3,

3π

4,

5π

6

I Join αi and αj by 0, 1, 2, 3 lines if the angle between αi andαj is π

2 ,2π3 ,

3π4 ,

5π6 , respectively

I If |αi | > |αj | and αi and αj are joined by a line then draw anarrow pointing from αi to αj

Σ of type Ar or Br

��� �� ��� �� ��� �� ��� ��α1 α2 αr−1 αr

I M = SLr+1(R)/SOr+1: (1, . . . , 1)

I M = SLr+1(C)/SUr+1: (2, . . . , 2)

I M = SLr+1(H)/Spr+1: (4, . . . , 4)

I M = E−266 /F4: (8, 8)

I M = SOo1,n+1/SOn+1 = RHn+1: (n), n ≥ 2

��� �� ��� �� ��� �� ��� �� ��� ��α1 α2 αr−2 αr−1 αr

+3

I M = SO2r+1(C)/SO2r+1: (2, . . . , 2, 2)

I M = SOor ,r+n/SOr × SOr+n: (1, . . . , 1, n), n ≥ 1

Iwasawa decomposition

I n =⊕

λ∈Σ+ gλ nilpotent subalgebra of g

I s = a⊕ n solvable subalgebra of g with [s, s] = n

I g = k⊕ a⊕ n Iwasawa decomposition of g (vector space directsum)

I G = KAN Iwasawa decomposition of G (diffeomorphism)

I M = G/K = AN = S solvable Lie group equipped withleft-invariant Riemannian metric

The setup for cohomogeneity one actions

I M = G/K connected irreducible Riemannian symmetric spaceof noncompact typeG noncompact semisimple real Lie groupK maximal compact subgroup of Go ∈ M with K · o = o

I H connected closed subgroup of G acting on M withcohomogeneity one

I L connected proper maximal subgroup of G with H ⊂ L

I g, k, h, l corresponding Lie algebras

Mostow 1961: l is reductive or parabolic

The reductive case

Karpelevich 1953: L has a totally geodesic orbit S $ M

=⇒ H has a totally geodesic orbit S

S reflectivem

geodesic reflection of M in S is an isometrym

∃ totally geodesic submanifold S⊥ of Mwith o ∈ S⊥ and ToS⊥ = νoS

Leung 1979: Classification of reflective submanifolds in irreduciblesimply connected Riemannian symmetric spaces

The reductive case (II)

Berndt-Tamaru 2004: S is a totally geodesic singular orbit of acohomogeneity one action on M ⇐⇒

I S reflective and rankS⊥ = 1, or

I S is one of the following totally geodesic non-reflectivesubmanifolds:

S M dim S dim M

CH2 G 22 /SO4 4 8

SL3(R)/SO3 G 22 /SO4 5 8

G 22 /SO4 SOo

3,4/SO3SO4 8 12

SL3(C)/SU3 G C2 /G2 8 14

G C2 /G2 SOC

7 /SO7 14 21

Parabolic subalgebras

I g = k⊕ p Cartan decomposition

I a maximal abelian subspace of p

I restricted root space decomposition

g = g0 ⊕

(⊕α∈Σ

gα

)

I Λ set of simple roots for Σ

I Φ subset of Λ, ΣΦ = Σ ∩ span{Φ}I lΦ = g0 ⊕

(⊕α∈ΣΦ

gα

), nΦ =

⊕α∈Σ+\Σ+

Φgα

lΦ reductive subalgebra, nΦ nilpotent subalgebra

I qΦ = lΦ ⊕ nΦ parabolic subalgebra; [qΦ, nΦ] ⊂ nΦ

(Chevalley decomposition)

I Every parabolic subalgebra of g is conjugate to qΦ for somesubset Φ ⊂ Λ

Parabolic subalgebras (II)

I aΦ = ∩α∈Φkerα , mΦ = lΦ aΦ

mΦ reductive subalgebra, aΦ abelian subalgebra

I qΦ = mΦ ⊕ aΦ ⊕ nΦ (Langlands decomposition)[qΦ, aΦ ⊕ nΦ] ⊂ aΦ ⊕ nΦ

I kΦ = k ∩ qΦ = k ∩mΦ

[kΦ,mΦ] ⊂ mΦ, [kΦ, aΦ] = {0}, [kΦ, nΦ] ⊂ nΦ

I F sΦ = MΦ · o semisimple symmetric space with rank equal to|Φ|, totally geodesic in M, boundary component of M w.r.t.maximal Satake compactification

I AΦ · o = Er−|Φ| Euclidean space, totally geodesic in M

I LΦ · o = FΦ = F sΦ × Er−|Φ| totally geodesic in M

I M = F sΦ × Er−|Φ| × NΦ (horospherical decomposition)

The boundary components

I Λ = {α1, . . . , αr}, {H1, . . . ,H r} dual basis of Λ in a

I HΦ =∑

αi∈Λ\Φ H i ∈ a ⊂ p ∼= ToM

I γΦ geodesic in M tangent to HΦ

I FΦ = {p ∈ M | p lies on geodesic parallel to γΦ}I F s

Φ = semisimple part of FΦ

I Er−|Φ| = Euclidean part of FΦ

Construction Method I: νo(H · o) ⊂ mΦ, Φ 6= ∅

I HsΦ ⊂ I o(F s

Φ) ⊂ MΦ acting on F sΦ with cohomogeneity one

I h = hsΦ ⊕ aΦ ⊕ nΦ subalgebra of qΦ

H acts on M with cohomogeneity one

Rank reduction - Such a cohomogeneity one action can beconstructed by a canonical extension of acohomogeneity one action on a boundary component

Construction Method II: νo(H · o) ⊂ aΦ

I Φ = ∅: qΦ = k0 ⊕ a⊕ n minimal parabolic subalgebra

I M = AN solvable Lie group with left-invariant metric

I ` ⊂ a one-dimensional linear subspace

I s` = (a `)⊕ n subalgebra of a⊕ n

S` acts on M with cohomogeneity one

Construction method II produces precisely all horospherefoliations on M

Construction Method III: νo(H · o) ⊂ nΦ

I Λ = {α1, . . . , αr}, {H1, . . . ,H r} dual basis of Λ in a

I Φj = Λ \ {αj}: Put qj = qΦj, nj = nΦj

, etcetera

I nνj =⊕

α∈Σ+\Σ+j ,α(H j )=ν gα

I nj =⊕

ν>0 nνj gradation generated by n1j

Assume that

I v ⊂ n1j ; define nj ,v = nj v subalgebra of nj

I NoLj

(nj ,v) = θNoLj

(v) acts transitively on Fj = F sj × E

I NoLj∩K (v) acts transitively on the unit sphere in v if dim v ≥ 2

Then

Hj ,v = NoLj

(nj ,v)Nj ,v acts on M with cohomogeneity one

Remark: dim v = 1 corresponds to foliation

Construction Method III - rank M = 1

M G K K0 gα n

RHn SOo1,n SOn SOn−1 Rn−1 Rn−1

CHn SU1,n Un Un−1 Cn−1 Cn−1 ⊕ RHHn Sp1,n Sp1Spn Sp1Spn−1 Hn−1 Hn−1 ⊕ R3

OH2 F−204 Spin9 Spin7 O O⊕ R7

I Λ = {α}, Φ = ∅, lΦ = g0 = k0 ⊕ a, nΦ = n = gα ⊕ g2α

I qΦ = g0 ⊕ n = k0 ⊕ a⊕ n minimal parabolic subalgebra

Problem: Find all k-dimensional (k ≥ 2) linear subspaces v of gαfor which there exists a subgroup of K0 acting transitively on theunit sphere in v

Construction Method III - rank M = 1

M G K K0 gα n

RHn SOo1,n SOn SOn−1 Rn−1 Rn−1

CHn SU1,n Un Un−1 Cn−1 Cn−1 ⊕ RHHn Sp1,n Sp1Spn Sp1Spn−1 Hn−1 Hn−1 ⊕ R3

OH2 F−204 Spin9 Spin7 O O⊕ R7

Berndt-Tamaru 2007:

I R: any linear subspace v ⊂ Rn−1

I C: any linear subspace v ⊂ Cn−1 with constant Kahler angle

I O: any linear subspace v ⊂ O of dimension k ∈ {2, 3, 4, 6, 7}I H: some linear subspaces v ⊂ Hn−1 with constant

quaternionic Kahler angle (no complete classification)

Constant quaternionic Kahler angle

Examples of subspaces V of Hn−1 with constant quaternionicKahler angle Φ:

I Φ = (0, 0, 0) V quaternionic

I Φ = (0, 0, π/2) V = Im(H)v , v ∈ Hn−1

I Φ = (0, π/2, π/2) V totally complex

I Φ = (π/2, π/2, π/2) V totally real

I Φ = (ϕ, π/2, π/2) V subspace with constant Kahler angle

ϕ ∈ (0, π/2) in a totally complex subspace of Hn−1

I Φ = (0, ϕ, ϕ) V complexification of a subspace with

constant Kahler angle ϕ ∈ (0, π/2) in a totally complexsubspace of Hn−1

The Classification

Berndt-Tamaru 2008: Let M be a connected irreducibleRiemannian symmetric space of noncompact type. Then everycohomogeneity one action on M either has a totally geodesicsingular orbit, or it is orbit equivalent to a cohomogeneity oneaction obtained by construction method I, II or III.

Cohomogeneity one actions on SL3(R)/SO3

I The boundary components of SL3(R)/SO3 areF s

1∼= F s

2∼= RH2

I There are three types of cohomogeneity one actions onRH2 = SL2(R)/SO2. Write SL2(R) = KAN.

I K = SO2: the canonical extension is a cohomogeneity oneaction with a minimal RH3 as a singular orbit

I N: the canonical extension leads to a horosphere foliation(with a singular point at infinity)

I A: the canonical extension leads to a foliation with a minimalhomogeneous hypersurface P as a leaf, where P is thecanonical extension of a geodesic in RH2

I RH2 × E ∼= F1∼= F2 is the only reflective submanifold S in

SL3(R)/SO3 for which S⊥ has rank one

Cohomogeneity one actions on SL3(R)/SO3 (II)

Theorem: Every cohomogeneity one action on SL3(R)/SO3 hasone of the following orbit structures:

I the reflective submanifold RH2 × E and the tubes around it

I a horosphere foliation

I the foliation given by the canonical extension of a geodesic inRH2 to a minimal homogeneous hypersurface, and itsequidistant hypersurfaces

I the minimal RH3 and the tubes around it

The case of foliations

I MF = set of all homogeneous codimension one foliations onM up to isometric congruence

I r = rank of M

I Aut(DD) ∈ {I ,Z2,S3} automorphism group of the Dynkindiagram associated to M

Berndt-Tamaru 2003: MF∼= (RP r−1 ∪ {1, . . . , r})/Aut(DD)

The two foliations on hyperbolic spaces

I horosphere foliationI foliation with exactly one minimal leaf S

I M = RHn: S = RHn−1 totally geodesicI M = CHn: S = ruled real hypersurface associated to a

horocycle in a totally geodesic RH2 ⊂ CHn

Duality and triality

MF depends only on the rank and on possible duality ortriality principles on the symmetric space

Example: r = 8, Aut(DD) = I

MF = RP7 ∪ {1, . . . , 8}

for the following symmetric spaces:

SOC17/SO17 ,SpR

8 /U8 , SpC8 /Sp8 , SOH

16/U16 , SOH17/U17

E 88 /SO16 , E C

8 /E8

and for the hyperbolic Grassmannians

G ∗8 (Rn+16) (n ≥ 1) , G ∗8 (Cn+16) (n ≥ 0) , G ∗8 (Hn+16) (n ≥ 0)

Geometry of the foliations

I ` ∈ RP r−1 ! horosphere foliation F` on M

I all leaves of F` are congruent to each other

I if r ≥ 2 then some foliations F` are harmonic (i.e. all leavesare minimal) and hence induce a harmonic map M → R

I i ∈ {1, . . . , r} ! αi ∈ Λ ! foliation Fi on M

I Fi contains exactly one minimal leaf

I |αi | = |αj | ⇒ Fi ,Fj have the same principal curvatures withthe same multiplicities

Corollary: If r ≥ 3, then there exist noncongruent homogeneousisoparametric systems on M with the same spectral data for thesecond fundamental form

Example

M = SL4(R)/SO4 =

x11 x12 x13 x14

0 x22 x23 x24

0 0 x33 x34

0 0 0 x44

xij ∈ R , x11x22x33x44 = 1

DD = ��� �� ��� �� ��� ��α1 α2 α3

F1 ≈

x11 0 x13 x14

0 x22 x23 x24

0 0 x33 x34

0 0 0 x44

, F2 ≈

x11 x12 x13 x14

0 x22 0 x24

0 0 x33 x34

0 0 0 x44

Outline of proof I

Step 1 There exists a connected closed solvable subgroup of Hwhich acts simply transitively on each orbit of H

Therefore may assume that H is solvable and acts simplytransitively on each orbit

Step 2 h ⊂ t⊕ a⊕ n for some Iwasawa decompositiong = k⊕ a⊕ n, where t is a maximal abelian subalgebra of zk(a)

Step 3 ha⊕n is a subalgebra of a⊕ n

Outline of proof II

Step 4 Classify the codimension one subalgebras of a⊕ n:

(a⊕ n) Rξ

with ξ ∈ a or ξ ∈ RHα ⊕ gα, α simple root

Step 5 Investigate the orbit equivalence of the correspondingcohomogeneity one actions:Write ha⊕n = (a⊕ n) Rξ with ξ ∈ a⊕ n

I ξ ∈ a⇒ h, ha⊕n induce same foliation

I ξ ∈ gα ⇒ h, ha⊕n induce same foliation

I ξ ∈ RHα ⊕ gα, ξ /∈ RHα, ξ /∈ gα ⇒ h, ha⊕n induce orbitequivalent foliations

Step 6 Investigate orbit equivalence of the model foliations F`,` ∈ RP r−1, and Fi , i ∈ {1, . . . , r}

Classification of homogeneous hypersurfaces in CHn

Berndt-Tamaru 2005: Every homogeneous real hypersurface inCHn is congruent to one of the following real hypersurfaces:

I a horosphere in CHn

I a ruled real hypersurface generated by a horocycle inRH2 ⊂ CHn, or to one of its equidistant hypersurfaces

I a tube around CHk ⊂ CHn for some k ∈ {0, . . . , n − 1}I a tube around RHn ⊂ CHn

I a tube around Fk ⊂ CHn for some k ∈ {2, . . . , n − 1}I a tube around Fk,ϕ ⊂ CHn for some k ∈ {1, . . . , [(n − 1)/2]}

and ϕ ∈ (0, π/2)

The submanifolds Fk and Fk,ϕ

I Iwasawa decomposition

su1,n = k⊕ a⊕ n ∼= un ⊕ a⊕ (gα ⊕ g2α)

I n = gα ⊕ g2α = (2n − 1)-dimensional Heisenberg algebra

I a ∼= R , g2α = Ja , gα ∼= Cn−1

I v linear subspace of gα ∼= Cn−1

=⇒ sv = a⊕ (gα v)⊕ g2α subalgebra of a⊕ n

=⇒ Sv · o homogeneous submanifold

I v k-dimensional real subspace of gα ∼= Cn−1, 2 ≤ k ≤ n − 1=⇒ Fk = Sv · o is a (2n − k)-dimensional homogeneoussubmanifold with real normal bundle of rank k

I v 2k-dimensional subspace of gα ∼= Cn−1 with constantKahler angle ϕ ∈ (0, π/2), 1 ≤ k ≤ [(n − 1)/2]=⇒ Fk,ϕ = Sv · o is a 2(n − k)-dimensional homogeneoussubmanifold with normal bundle of rank 2k and constantKahler angle ϕ

Constant principal curvatures in CHn

Assume that M ⊂ CHn is real hypersurface with constant principalcurvatures, and define g = number of distinct principal curvatures

M homogeneous =⇒ g ∈ {2, 3, 4, 5}

Problems:

I Does g ∈ {2, 3, 4, 5} hold in general?Note: g = 1 (umbilical) is impossible

I Is any real hypersurface of CHn with constant principalcurvatures an open part of a homogeneous hypersurface?

I YES for g = 2 (Montiel 1985)I YES for Hopf hypersurfaces (Berndt 1989)

Classification of real hypersurfaces with g = 3 in CHn

(Berndt–Dıaz-Ramos 2006, 2007): Every real hypersurface inCHn (n ≥ 2) with three distinct constant principal curvatures is anopen part of a homogeneous hypersurface:

I a ruled real hypersurface generated by a horocycle inRH2 ⊂ CHn, or to one of its equidistant hypersurfaces

I a tube around CHk ⊂ CHn for some k ∈ {1, . . . , n − 2}I a tube with radius r 6= ln(2 +

√3) around RHn ⊂ CHn

I a tube with radius r = ln(2 +√

3) around Fk ⊂ CHn for somek ∈ {2, . . . , n − 1}.

Outline of proof (I)

I λ1, λ2, λ3 principal curvatures

I m1,m2,m3 corresponding multiplicities

I Tλ1 ,Tλ2 ,Tλ3 corresponding eigenbundles

TM = Tλ1 ⊕ Tλ2 ⊕ Tλ3

I ξ (local) unit normal vector field (Jξ is Hopf vector field onM)

We can assume that M is non-Hopf: RJξ * Tλi

Step 1 There exists no point p ∈ M such that the orthogonalprojections of Jξp onto Tλi

(p), i = 1, 2, 3, are nontrivial.

Thus we can assume that there exists a point p ∈ M such thatJξp = b1u1 + b2u2 with some unit vectors ui ∈ Tλi

(p) and0 6= bi ∈ R, i = 1, 2.

Outline of proof (II)

Step 2 There exists a unit vector a ∈ Tλ3(p) such thatRξp ⊕ Ru1 ⊕ Ru2 ⊕ Ra is a complex subspace of TpCHn.

Must hold on an open neighborhood of p−→ vector fields U1,U2,A near p

Remark: The integral curves of A are geodesics in M and the threevector fields A,U1,U2 span an autoparallel distribution

Step 3 mi > 1 =⇒ 4λ3λi = 1 (i ∈ {1, 2})

=⇒ m1 = 1 or m2 = 1

Thus we can assume m2 = 1

Outline of proof (III)

Step 4

Case 1: m1 > 1

λ1 =√

3/2, λ2 = 0, λ3 =√

3/6Tλ1 RU1 is real

J(Tλ1 RU1) ⊂ Tλ3

Case 2: m1 = 1

−1/2 < λ3 < 1/2

λ1,2 = 12

(3λ3 ∓

√1− 3λ2

3

)

Outline of proof (IV)

Step 5Investigate focal sets and equidistant hypersurfaces using Jacobifield theory:There exists a (2n −m1)-dimensional focal manifold (if m1 > 1) orequidistant hypersurface (if m1 = 1) F of M with totally realnormal bundle νF , and a unit vector field Z tangent to themaximal holomorphic subbundle of TM such that the secondfundamental form α of F is given by the trivial bilinear extension of

2α(Z , Jξ) = ξ

for all ξ ∈ Γ(νF ).

Outline of proof (V)

Step 6Prove the following rigidity result:

Let F be a (2n − k)-dimensional connected submanifold in CHn,n ≥ 3, with totally real normal bundle νF . Assume that thereexists a unit vector field Z tangent to the maximal holomorphicsubbundle of TF such that the second fundamental form α of F isgiven by the trivial bilinear extension of 2α(Z , Jξ) = ξ for allξ ∈ Γ(νF ). Then F is holomorphically congruent to an open partof the ruled minimal submanifold Fk .

k = 1: F1 is the homogeneous ruled real hypersurface determinedby a horocycle in RH2 ⊂ CHn

Open problems

I Let M be a real hypersurface in CHn with g distinct constantprincipal curvatures. Is it true that g ∈ {2, 3, 4, 5}?

I Classification of real hypersurfaces in CHn with 4 or 5 distinctconstant principal curvatures

I Is every real hypersurface in CHn with constant principalcurvatures an open part of a homogeneous hypersurface?

Polar actions and hyperpolar actions

Let M be a connected complete Riemannian manifold and H be aconnected closed subgroup of I (M)

The action of H on M is said to be polar if there exists aconnected complete submanifold S of M such that

I ∀p ∈ M : (H · p) ∩ S 6= ∅I ∀p ∈ S : TpS ⊂ νp(H · p)

Any such submanifold S is called a section of the action

A polar action on M is called hyperpolar if there exists a flatsection.

Every section of a polar action is totally geodesic

Problem. Classify polar/hyperpolar actions onsymmetric spaces

Polar/hyperpolar actions - the compact case

M irreducible simply connected symmetric space of compact type

I Dadok 1985: Classification of polar actions on spheres

I Podesta-Thorbergsson 1999: Classification of polar actionson projective spaces

I Kollross 2001, 2007: Classification of hyperpolar actions forhigher rank; every polar action is hyperpolar

Polar/hyperpolar actions - the noncompact case

I M = G/K symmetric space of noncompact type

I F foliation on M

I F polar (resp. hyperpolar) if F has a section (resp. a flatsection)

Example. Consider the Chevalley decomposition qΦ = lΦ ⊕ nΦ ofa parabolic subalgebra qΦ of g. Then the orbits of NΦ form a polarfoliation on M which is hyperpolar if and only if Φ = ∅. Thetotally geodesic submanifold LΦ · o is a section.

Examples of hyperpolar foliations

I V linear subspace Em

=⇒ FmV = {p + V | p ∈ Em} homogeneous hyperpolar

foliation on Em

I F ∈ {R,C,H,O}, M = G/K = FHn

g = k⊕ a⊕ n Iwasawa decompositionn = gα ⊕ g2α

` ⊂ gα one-dimensional linear subspaces` = a⊕ (n `) subalgebra of a⊕ n

=⇒ FnF = {S` · p | p ∈ FHn} homogeneous hyperpolar

foliation on FHn

I Fn1F1× · · · × Fnk

Fk×Fm

V homogeneous hyperpolar foliation onF1Hn1 × · × FkHnk × Em

Examples of hyperpolar foliations (II)

I M = G/K symmetric space of noncompact type

I qΦ = mΦ ⊕ aΦ ⊕ nΦ Langlands decomposition of parabolicsubalgebra of g

I Φ orthogonal set of roots, that is, ∀α, β ∈ Φ : 〈α, β〉 = 0

I FΦ∼= F1Hn1 × · × F|Φ|Hn|Φ| × Er−|Φ|

I Fn1F1× · · · × Fn|Φ|

F|Φ|×F r−rΦ

V homogeneous hyperpolar foliation

on FΦ

I FΦ,V = Fn1F1× · · · × Fn|Φ|

F|Φ|×F r−rΦ

V × NΦ homogeneous

hyperpolar foliation on M = FΦ × NΦ

I F∅,{0} horocycle foliation on M

Classification of hyperpolar foliations on symmetricspaces of noncompact type

Berndt-DiazRamos-Tamaru 2008: Let M be a symmetric spaceof noncompact type. Every homogeneous hyperpolar foliation onM is isometrically congruent to FΦ,V for some orthogonal set Φ ofsimple roots and some linear subspace V ⊂ Er−|Φ|.

Open problems

For symmetric spaces of noncompact type:

I Is every hyperpolar foliation of codimension ≥ 2homogeneous?

I Find examples of polar/hyperpolar actions with singular orbits

I Find examples of homogeneous polar foliations

I Classify polar foliations

I Classify polar/hyperpolar actions with singular orbits