h-type sub-riemannian manifolds -...

H-type sub-Riemannian manifolds

Fabrice Baudoin

Based on a joint work with E. Grong, G. Molino & L. Rizzi

Sub-Riemannian manifolds

Let M be a smooth, connected manifold with dimension n + m.

Weassume that M is equipped with a bracket generating sub-bundleH ⊂ TM of rank n and a fiberwise inner product gH on thatdistribution.

The distribution H is referred to as the set of horizontaldirections.Sub-Riemannian geometry is the study of the geometry whichis intrinsically associated to (H, gH).


Let M be a smooth, connected manifold with dimension n + m. Weassume that M is equipped with a bracket generating sub-bundleH ⊂ TM of rank n and a fiberwise inner product gH on thatdistribution.




The distribution H is referred to as the set of horizontaldirections.

Sub-Riemannian geometry is the study of the geometry whichis intrinsically associated to (H, gH).

Example: H-type groups

Let g be a finite-dimensional Lie algebra with non-trivial center z.We say that g is of H-type if g is equipped with an inner product〈·, ·〉 such that :

1 [z⊥, z⊥] = z;2 For each z ∈ z, define Jz : z⊥ → z⊥ by

〈Jzx , y〉 = 〈z , [x , y ]〉 .

Then, Jz is an orthogonal map when ‖z‖ = 1.An H-type group is a connected, simply connected Lie group whoseLie algebra is of H-type.

Example: Heisenberg group



1 [z⊥, z⊥] = z;

2 For each z ∈ z, define Jz : z⊥ → z⊥ by

〈Jzx , y〉 = 〈z , [x , y ]〉 .






〈Jzx , y〉 = 〈z , [x , y ]〉 .

Then, Jz is an orthogonal map when ‖z‖ = 1.

An H-type group is a connected, simply connected Lie group whoseLie algebra is of H-type.





〈Jzx , y〉 = 〈z , [x , y ]〉 .




H-type groups yield an interesting class of sub-Riemannianmanifolds.

Indeed, given an H-type group H we define thehorizontal distribution H as the left-invariant distribution on Hwhose fiber at the identity is z⊥. Denoting g the left-invariantRiemannian metric on H induced by 〈·, ·〉, then (H,H, g|H) is a(step 2) sub-Riemannian manifold.


H-type groups yield an interesting class of sub-Riemannianmanifolds. Indeed, given an H-type group H we define thehorizontal distribution H as the left-invariant distribution on Hwhose fiber at the identity is z⊥.

Denoting g the left-invariantRiemannian metric on H induced by 〈·, ·〉, then (H,H, g|H) is a(step 2) sub-Riemannian manifold.


H-type groups yield an interesting class of sub-Riemannianmanifolds. Indeed, given an H-type group H we define thehorizontal distribution H as the left-invariant distribution on Hwhose fiber at the identity is z⊥. Denoting g the left-invariantRiemannian metric on H induced by 〈·, ·〉, then (H,H, g|H) is a(step 2) sub-Riemannian manifold.


In H-type groups, the sub-Riemannian structure comes with twoadditional structures that we can take advantage of to study thesub-Riemannian intrinsic geometry:

A totally geodesic Riemannian foliation structure given by theorthogonal bundle H⊥.At each point p, there is a representation of the Cliffordalgebra Cl(H⊥p ) onto the space of orthogonal maps Hp → Hp.



A totally geodesic Riemannian foliation structure given by theorthogonal bundle H⊥.

At each point p, there is a representation of the Cliffordalgebra Cl(H⊥p ) onto the space of orthogonal maps Hp → Hp.



A totally geodesic Riemannian foliation structure given by theorthogonal bundle H⊥.At each point p, there is a representation of the Cliffordalgebra Cl(H⊥p ) onto the space of orthogonal maps Hp → Hp.

H-type sub-Riemannian spaces

Our main (rough) question in this talk: Classify and study allsub-Riemannian spaces that carry those two additional structures.

Such space will be called an H-type sub-Riemannian space. Somefeatures of the class of H-type sub-Riemannian spaces:

1) The models of H-type sub-Riemannian geometry are completelyintegrable meaning that explicit computations are possible: Exactformulas for the sub-Riemannian heat kernels, Exact formulas forsolutions of the Jacobi equation, Exact formulas for the volume ofsub-Riemannian balls and sub-Riemannian diameter, Exact formulasfor the sub-Riemannian spectrum, etc... H-type groups are the flatmodels.



Such space will be called an H-type sub-Riemannian space.

Somefeatures of the class of H-type sub-Riemannian spaces:





1) The models of H-type sub-Riemannian geometry are completelyintegrable meaning that explicit computations are possible: Exactformulas for the sub-Riemannian heat kernels

, Exact formulas forsolutions of the Jacobi equation, Exact formulas for the volume ofsub-Riemannian balls and sub-Riemannian diameter, Exact formulasfor the sub-Riemannian spectrum, etc... H-type groups are the flatmodels.




1) The models of H-type sub-Riemannian geometry are completelyintegrable meaning that explicit computations are possible: Exactformulas for the sub-Riemannian heat kernels, Exact formulas forsolutions of the Jacobi equation

, Exact formulas for the volume ofsub-Riemannian balls and sub-Riemannian diameter, Exact formulasfor the sub-Riemannian spectrum, etc... H-type groups are the flatmodels.




1) The models of H-type sub-Riemannian geometry are completelyintegrable meaning that explicit computations are possible: Exactformulas for the sub-Riemannian heat kernels, Exact formulas forsolutions of the Jacobi equation, Exact formulas for the volume ofsub-Riemannian balls and sub-Riemannian diameter,

Exact formulasfor the sub-Riemannian spectrum, etc... H-type groups are the flatmodels.




1) The models of H-type sub-Riemannian geometry are completelyintegrable meaning that explicit computations are possible: Exactformulas for the sub-Riemannian heat kernels, Exact formulas forsolutions of the Jacobi equation, Exact formulas for the volume ofsub-Riemannian balls and sub-Riemannian diameter, Exact formulasfor the sub-Riemannian spectrum, etc...

H-type groups are the flatmodels.


2) On a H-type sub-Riemannian manifold, there is a canonicalconnection.

One can compute from this connection tensorialcurvature quantities that control comparison with respect to themodel spaces.

As a consequence, the program of geometric analysis is ratheradvanced in the class of H-type sub-Riemannian spaces: BonnetMyers type theorems, Sub-Laplacian comparison theorems , Volumecomparison theorems , Li-Yau subelliptic heat kernel estimates,Sobolev and isoperimetric inequalities, ...


2) On a H-type sub-Riemannian manifold, there is a canonicalconnection. One can compute from this connection tensorialcurvature quantities that control comparison with respect to themodel spaces.




As a consequence, the program of geometric analysis is ratheradvanced in the class of H-type sub-Riemannian spaces: BonnetMyers type theorems

, Sub-Laplacian comparison theorems , Volumecomparison theorems , Li-Yau subelliptic heat kernel estimates,Sobolev and isoperimetric inequalities, ...



As a consequence, the program of geometric analysis is ratheradvanced in the class of H-type sub-Riemannian spaces: BonnetMyers type theorems, Sub-Laplacian comparison theorems

, Volumecomparison theorems , Li-Yau subelliptic heat kernel estimates,Sobolev and isoperimetric inequalities, ...



As a consequence, the program of geometric analysis is ratheradvanced in the class of H-type sub-Riemannian spaces: BonnetMyers type theorems, Sub-Laplacian comparison theorems , Volumecomparison theorems

, Li-Yau subelliptic heat kernel estimates,Sobolev and isoperimetric inequalities, ...



As a consequence, the program of geometric analysis is ratheradvanced in the class of H-type sub-Riemannian spaces: BonnetMyers type theorems, Sub-Laplacian comparison theorems , Volumecomparison theorems , Li-Yau subelliptic heat kernel estimates

,Sobolev and isoperimetric inequalities, ...



As a consequence, the program of geometric analysis is ratheradvanced in the class of H-type sub-Riemannian spaces: BonnetMyers type theorems, Sub-Laplacian comparison theorems , Volumecomparison theorems , Li-Yau subelliptic heat kernel estimates,Sobolev and isoperimetric inequalities

, ...

Bundle-like totally geodesic foliations

Let (M, g) be a Riemannian manifold and F be a Riemannianfoliation on M which is bundle-like and totally geodesic.

Thismeans that we have an orthogonal splitting of the tangent bundle:

TM = H⊕ V

where the sub-bundle V is integrable. For X ∈ Γ(H), Z ∈ Γ(V)

LZg(X ,X ) = 0 (bundle like property) ;LXg(Z ,Z ) = 0 (totally geodesic leaves).


Let (M, g) be a Riemannian manifold and F be a Riemannianfoliation on M which is bundle-like and totally geodesic. Thismeans that we have an orthogonal splitting of the tangent bundle:

TM = H⊕ V

where the sub-bundle V is integrable.

For X ∈ Γ(H), Z ∈ Γ(V)




TM = H⊕ V


LZg(X ,X ) = 0 (bundle like property)

;LXg(Z ,Z ) = 0 (totally geodesic leaves).



TM = H⊕ V



Sub-Riemannian structures arising from foliations

Examples:

The Hopf fibration S1 → S2n+1 → CPn induces asub-Riemannian structure on S2n+1 which comes from atotally geodesic foliation.

The pseudo-Riemannian anti de-Sitter submersionAdS2n+1 → CHn induces a sub-Riemannian structure onAdS2n+1 which comes from a totally geodesic foliation.More generally, totally geodesic Riemannian submersions,Sasakian and 3-Sasakian manifolds provide examples ofsub-Riemannian structures associated with totally geodesicfoliations.H-type groups.


Examples:

The Hopf fibration S1 → S2n+1 → CPn induces asub-Riemannian structure on S2n+1 which comes from atotally geodesic foliation.The pseudo-Riemannian anti de-Sitter submersionAdS2n+1 → CHn induces a sub-Riemannian structure onAdS2n+1 which comes from a totally geodesic foliation.

More generally, totally geodesic Riemannian submersions,Sasakian and 3-Sasakian manifolds provide examples ofsub-Riemannian structures associated with totally geodesicfoliations.H-type groups.


Examples:

The Hopf fibration S1 → S2n+1 → CPn induces asub-Riemannian structure on S2n+1 which comes from atotally geodesic foliation.The pseudo-Riemannian anti de-Sitter submersionAdS2n+1 → CHn induces a sub-Riemannian structure onAdS2n+1 which comes from a totally geodesic foliation.More generally, totally geodesic Riemannian submersions,Sasakian and 3-Sasakian manifolds provide examples ofsub-Riemannian structures associated with totally geodesicfoliations.

H-type groups.


Examples:

The Hopf fibration S1 → S2n+1 → CPn induces asub-Riemannian structure on S2n+1 which comes from atotally geodesic foliation.The pseudo-Riemannian anti de-Sitter submersionAdS2n+1 → CHn induces a sub-Riemannian structure onAdS2n+1 which comes from a totally geodesic foliation.More generally, totally geodesic Riemannian submersions,Sasakian and 3-Sasakian manifolds provide examples ofsub-Riemannian structures associated with totally geodesicfoliations.H-type groups.

The canonical connection

Theorem

There exists a unique metric connection ∇ on M, called the Bottconnection of the foliation, such that:

H and V are ∇-parallel, i.e. for every X ∈ Γ(H), Y ∈ Γ(M)and Z ∈ Γ(V),

∇YX ∈ Γ(H), ∇YZ ∈ Γ(V);

The torsion T of ∇ satisfies

T (H,H) ⊂ V, T (H,V) = 0, T (V,V) = 0.

J-map

For Z ∈ Γ∞(V), there is a unique skew-symmetric fiberendomorphism JZ : Γ(H)→ Γ(H) such that for all horizontalvector fields X and Y ,

gH(JZX ,Y ) = gV(Z ,T (X ,Y )).

H-type foliations

Definition

We say that (M,H, g) is an H-type foliation if for every Z ∈ Γ(V)and X ,Y ∈ Γ(H), 〈JZX , JZY 〉 = ‖Z‖2〈X ,Y 〉.

If the horizontal divergence of the torsion of the Bottconnection is zero, then we say that (M,H, g) is an H-typefoliation of Yang-Mills type.If the torsion of the Bott connection is horizontally parallel, i.e.∇HT = 0, then we say that (M,H, g) is an H-type foliationwith horizontally parallel torsion.If the torsion of the Bott connection is completely parallel, i.e.∇T = 0, then we say that (M,H, g) is an H-type foliationwith parallel torsion.

H-type foliations

Definition


If the horizontal divergence of the torsion of the Bottconnection is zero, then we say that (M,H, g) is an H-typefoliation of Yang-Mills type.

If the torsion of the Bott connection is horizontally parallel, i.e.∇HT = 0, then we say that (M,H, g) is an H-type foliationwith horizontally parallel torsion.If the torsion of the Bott connection is completely parallel, i.e.∇T = 0, then we say that (M,H, g) is an H-type foliationwith parallel torsion.

H-type foliations

Definition


If the horizontal divergence of the torsion of the Bottconnection is zero, then we say that (M,H, g) is an H-typefoliation of Yang-Mills type.If the torsion of the Bott connection is horizontally parallel, i.e.∇HT = 0, then we say that (M,H, g) is an H-type foliationwith horizontally parallel torsion.

If the torsion of the Bott connection is completely parallel, i.e.∇T = 0, then we say that (M,H, g) is an H-type foliationwith parallel torsion.

H-type foliations

Definition


If the horizontal divergence of the torsion of the Bottconnection is zero, then we say that (M,H, g) is an H-typefoliation of Yang-Mills type.If the torsion of the Bott connection is horizontally parallel, i.e.∇HT = 0, then we say that (M,H, g) is an H-type foliationwith horizontally parallel torsion.If the torsion of the Bott connection is completely parallel, i.e.∇T = 0, then we say that (M,H, g) is an H-type foliationwith parallel torsion.

Structure Torsion ReferenceComplex Type, m = 1, n = 2kK-Contact YM [1] [4]Sasakian CP [1] [7]Heisenberg Group CP [9]Hopf Fibration S1 ↪→ S2k+1 → CPk CP [3]Anti de-Sitter Fibration S1 ↪→ AdS2k+1(C)→ CHk CP [8] [19]Twistor Type, m = 2, n = 4kTwistor space over quaternionic Kähler manifold HP [11] [17]Projective Twistor space CP1 ↪→ CP2k+1 → HPk HP [5]Hyperbolic Twistor space CP1 ↪→ CH2k+1 → HHk HP [2] [8]Quaternionic Type, m = 3, n = 4k3-K Contact YM [13] [18]Neg. 3-K Contact YM [13] [18]3-Sasakian HP [6] [16]Neg. 3-Sasakian HP [6]Torus bundle over hyperkähler manifolds CP [12]Quat. Heisenberg Group CP [9]Quat. Hopf Fibration SU(2) ↪→ S4k+3 → HPk HP [5]Quat. Anti de-Sitter Fibration SU(2) ↪→ AdS4k+3(H)→ HHk HP [2] [8]Octonionic Type, m = 7, n = 8Oct. Heisenberg Group CP [9]Oct. Hopf Fibration S7 ↪→ S15 → OP1 HP [15]Oct. Anti de-Sitter Fibration S7 ↪→ AdS15(O)→ OH1 HP [8]H-type Groups, m is arbitrary CP [10] [14]

Table: Some examples of H-type foliations.

H-type foliations are Yang-Mills

Although H-type foliations are not necessarily horizontally parallel,they are always Yang-Mills.

Theorem

Let (M, g ,H) be an H-type foliation. Then it satisfies theYang-Mills condition.

Sub-Laplacian

Let (M, g ,H) be an H-type foliation and assume the Riemannianmetric g to be complete.The horizontal Laplacian ∆H of the foliation is the generator of thesymmetric closable bilinear form in L2(M, µg ):

EH(u, v) =

∫M〈∇Hu,∇Hv〉 dµg , u, v ∈ C∞0 (M).

The H-type property implies that H is bracket generating (it isactually fat). The differential operator ∆H is not elliptic but it ishypoelliptic.

Sub-Laplacian


EH(u, v) =

∫M〈∇Hu,∇Hv〉 dµg , u, v ∈ C∞0 (M).

The H-type property implies that H is bracket generating (it isactually fat).

The differential operator ∆H is not elliptic but it ishypoelliptic.

Sub-Laplacian


EH(u, v) =

∫M〈∇Hu,∇Hv〉 dµg , u, v ∈ C∞0 (M).

The H-type property implies that H is bracket generating (it isactually fat). The differential operator ∆H is not elliptic but it ishypoelliptic.

First eigenvalue estimate

Theorem

Let (M,H, g) be a complete H-type foliation with RicciH ≥ KgH,with K > 0. Then the first non zero eigenvalue of thesub-Laplacian −∆H satisfies

λ1 ≥nK

n + 3m − 1.

The estimate is sharp on the Hopf and the quaternionic Hopffibrations, since it is an equality on those model spaces.

Bonnet-Myers diameter estimate

Theorem

Let (M,H, g) be a complete H-type foliation with RicciH ≥ KgH,with K > 0. then M is compact with a finite fundamental groupand

diam(M, dCC ) ≤ 2√3π

√(n + 4m)(n + 6m)

nK.

The theorem only assumes a lower bound on the horizontal Riccicurvature (not pseudo-Hermitian type sectional curvature !), butthe estimate is not sharp in the models. The proof does not useJacobi fields or geodesics but is rather purely analytic and relies onSobolev type inequalities using generalized curvature dimensionestimates ( after B.-Garofalo).


Theorem



√(n + 4m)(n + 6m)

nK.

The theorem only assumes a lower bound on the horizontal Riccicurvature (not pseudo-Hermitian type sectional curvature !), butthe estimate is not sharp in the models.

The proof does not useJacobi fields or geodesics but is rather purely analytic and relies onSobolev type inequalities using generalized curvature dimensionestimates ( after B.-Garofalo).


Theorem



√(n + 4m)(n + 6m)

nK.

The theorem only assumes a lower bound on the horizontal Riccicurvature (not pseudo-Hermitian type sectional curvature !), butthe estimate is not sharp in the models. The proof does not useJacobi fields or geodesics but is rather purely analytic and relies onSobolev type inequalities using generalized curvature dimensionestimates ( after B.-Garofalo).

Parallel horizontal Clifford structures

At any point, we can identify the space ∧2V with the linearsubspace Cl2(V) ⊂ Cl(V) obtained through the canonicalisomorphism Z1 ∧ Z2 → Z1 · Z2 + 〈Z1,Z2〉.

Definition

Let (M, g ,H) be an H-type foliation with horizontally paralleltorsion. We say that (M, g ,H) is an H-type foliation with a parallelhorizontal Clifford structure if there exists a smooth bundle mapΨ : V × V → Cl2(V) such that for every Z1,Z2 ∈ V

(∇Z1J)Z2 = JΨ(Z1,Z2).


Theorem

Let (M, g ,H) be an H-type foliation with parallel horizontalClifford structure. Then, there exists a constant κ ∈ R such thatfor every u, v ∈ V

Ψ(u, v) = −κ(u · v + 〈u, v〉).

Moreover the sectional curvature of the leaves of the foliationassociated to V is constant equal to κ2. In particular, if the torsionis completely parallel, the leaves are flat.


Theorem


Ψ(u, v) = −κ(u · v + 〈u, v〉).

Moreover the sectional curvature of the leaves of the foliationassociated to V is constant equal to κ2.

In particular, if the torsionis completely parallel, the leaves are flat.


Theorem


Ψ(u, v) = −κ(u · v + 〈u, v〉).

Moreover the sectional curvature of the leaves of the foliationassociated to V is constant equal to κ2. In particular, if the torsionis completely parallel, the leaves are flat.

Parallel horizontal Clifford structures: κ = 0I

Theorem

Let π : (M, g)→ (B, h) be a Riemannian submersion with totallygeodesic fibers. Assume that B is simply connected and that(M,H, g) is an H-type foliation with completely parallel torsion,where H is the horizontal space of π. Then one of the following(non exclusive) cases occur:

m = 1 and B is Kähler;m = 2 or m = 3 and B is locally hyper-Kähler;m is arbitrary and B is flat, thus isometric to a representationof the Clifford algebra Cl(Rm).

Parallel horizontal Clifford structures: κ > 0

Parallel horizontal Clifford structures: κ < 0

Horizontal Einstein property

Theorem

Let (M, g ,H) be an H-type foliation with a parallel horizontalClifford structure with m ≥ 2 such that:

Ψ(u, v) = −κ(u · v + 〈u, v〉), u, v ∈ V,

with κ ∈ R,

then:If m 6= 3, RicciH = κ

(n4 + 2(m − 1)

)gH.

If m = 3 and (M, g ,H) is of quaternionic type thenRicciH = κ

(n2 + 4

)gH.

If m = 3 and (M, g ,H) is not of quaternionic type, then atany point, H orthogonally splits as a direct sum H+ ⊕H− andfor X ,Y ∈ H,

RicciH(X ,Y )

= κ(n4

+ 2(m − 1))〈X ,Y 〉+κ

4(dimH+−dimH−)〈σ(X ),Y 〉,

where σ = IdH+ ⊕ (−IdH−).

Horizontal Einstein property

Theorem

Let (M, g ,H) be an H-type foliation with a parallel horizontalClifford structure with m ≥ 2 such that:

Ψ(u, v) = −κ(u · v + 〈u, v〉), u, v ∈ V,

with κ ∈ R, then:If m 6= 3, RicciH = κ

(n4 + 2(m − 1)

)gH.

If m = 3 and (M, g ,H) is of quaternionic type thenRicciH = κ

(n2 + 4

)gH.

If m = 3 and (M, g ,H) is not of quaternionic type, then atany point, H orthogonally splits as a direct sum H+ ⊕H− andfor X ,Y ∈ H,

RicciH(X ,Y )

= κ(n4

+ 2(m − 1))〈X ,Y 〉+κ

4(dimH+−dimH−)〈σ(X ),Y 〉,

where σ = IdH+ ⊕ (−IdH−).

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h-type sub-riemannian manifolds -...

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