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Background H-type Foliations Clifford Structures Comparison Theorems End Notes

Comparison Theorems on H-type Foliations

Gianmarco MolinoFabrice Baudoin, Erlend Grong, and Luca Rizzi

Temple University

4 November, 2019


Riemannian Geometry

Recall that Riemannian geometry is the study of manifolds(generalized Euclidean spaces) that have an infinitesimal notion ofdistance. In particular, we have the following structures:

Topological: Notion of “nearness” for pointsManifold: Local homeomorphisms to Rn

Smooth: Derivatives of functions are well-definedRiemannian: Distance, and length are well-defined in alldirections


Connections

Denoting the space of vector fields X(M), an operator

∇ : X(M)× X(M) → X(M)

such that1 ∇fX+YZ = f∇XZ +∇YZ2 ∇X(fY) = df(X)Y + f∇XY

is called a connection.


CurvatureEquipped with a connection, we can define curvature:

Riemannian Curvature:

R(X,Y)Z = ∇X∇YZ −∇Y∇XZ −∇∇XY−∇YXZ

Sectional Curvature: (for orthonormal X,Y):

S(X,Y) = g(R(X,Y)Y,X)

Ricci Curvature:

Ric(X,Y) = Trg(Z 7→ R(Z,X)Y)

Scalar Curvature:K = Trg(Ric)


Levi-Civita Connection

While there are many possible connections on a manifold, there isone in particular that we are interested in.

TheoremThere exists a unique connection ∇ such that

1 ∇X(g(Y,Z)) = g(∇XY,Z) + g(Y,∇XZ)2 ∇XY −∇YX = [X,Y]

This connection is called the Levi-Civita connection.


Basic Definitions, sub-Riemannian Geometry

Let M be a smooth manifold. We say that(M,H, gH) is a sub-Riemannian manifold if

H is a constant rank, bracket generating subbundle of TM,and gH is a inner product on H.

(M,H, g) is a sub-Riemannian manifold with metricpreserving complement or sRmc-manifold if

(M, g) is a Riemannian manifold,the metric orthogonally splits as g = gH ⊕ gV ,and (M,H, gH) is a sub-Riemannian manifold.

We denote by V the orthogonal complement of H by g.


sub-Riemannian Geometry versus Riemannian Geometry

Riemannian geometry is distinguished by the Riemannianmetric g and thus a definition of distance in all directions.sub-Riemannian geometry has a notion of distance, but onlyin some directions.A major goal of sub-Riemannian geometry is to recoverRiemannian results (or generalizations).The Chow-Rashevskii theorem guarantees that if thehorizontal bundle is bracket generating, then any two pointsare connected by a horizontal path.


Riemannian Comparison TheoremOn a Riemannian manifold assume there exists κ such that

Ric(X,Y) ≥ (n − 1)κg(X,Y)

Laplacian (Rauch) Comparison Theorem:

∆r ≤

(n − 1)√κ cot(

√κr) κ > 0

n−1r κ = 0

(n − 1)√|κ| coth(

√|κ|r) κ < 0

Bonnet-Meyers Diameter Estimates: If κ > 0 then

diam(M) ≤ π√κ

and the fundamental group of M must be finite.


Motivating Example: Hopf Fibration

Consider S2n+1 foliated as

S1 ↪→ S2n+1 π−→ CPn

Defining the horizontal distribution as

H = dπ−1(TCPn)

the space (S2n+1,H, g) is an sRmc-manifold, foliated with leaves

Vx ∼= S1


Gromov-Hausdorff Convergence

For a sRmc-manfiold (M,H, g) we define the canonical variation ofthe metric

gε = gH +1ε

gV

which in the Gromov-Hausdorff sense

(M,H, gε) ε→0+−−−→ (M,H, gH)

The key idea of this project is to leverage our knowledge of theRiemannian structure to understand the sub-Riemannian one.


Hladky-Bott Connection

Theorem (Hladky ‘12 [5])There exists a unique metric connection ∇ on (M,H, g) such that

1 H and V are ∇-parallel,2 The torsion T of ∇ satisfies

T(H,H) ⊂ V,T(V,V) ⊂ H

3 For every X,Y ∈ Γ(H),Z,V ∈ Γ(V),⟨T(X,Z),Y⟩H = ⟨T(Y,Z),X⟩H⟨T(Z,X),V⟩V = ⟨T(V,X),Z⟩V .

This is called the Hladky-Bott connection.


Hladky-Bott Connection

We can explicitly write ∇ in terms of the Levi-Civita connection∇g as

∇XY =

πH∇g

XY X,Y ∈ Γ(H)

πH[X,Y] + AXY Y ∈ Γ(H),X ∈ Γ(V)πV [X,Y] + AXY Y ∈ Γ(V),X ∈ Γ(H)

πV∇gXY X,Y ∈ Γ(V)

where the tensor A is defined by

⟨AXY,Z⟩ = 12 ((LXVg)(YH,ZH) + (LXHg)(YV ,ZV))


Bundle-like Metrics and Totally Geodesic Foliations

There are two important properties we will require:Bundle-like metric: A foliation is said to have a bundle-likemetric if the metric locally splits orthogonally. This isequivalent to

LVg(H,H) = 0

Totally geodesic foliation: A foliation is said to be totallygeodesic if the geodesics of the fibers are embedded geodesicsof the total space. This is equivalent to

LHg(V,V) = 0


J Map

On (M,H, g) we can associate to each vector field Z ∈ Γ(TM) anendomorphism JZ of TM defined by

⟨JZX,Y⟩ = ⟨Z,T(X,Y)⟩

If V is integrable, {JZX ∈ H if Z ∈ V,X ∈ HJZX = 0 otherwise

We thus take the perspective

J : V → End(H), Z 7→ JZ


H-type Foliations

DefinitionLet (M,H, g) be a sRmc-manfiold. We say that (M,H, g, J) is anH-type foliation if

1 (M,V, g) is a totally geodesic foliation with bundle-like metric,2 V is integrable, and3 for all X,Y ∈ Γ(H),Z ∈ Γ(V),

⟨JZX, JZY⟩H = ∥Z∥2⟨X,Y⟩H


Parallel Torsion

We also refine the definition of H-type foliations based on thebehavior of derivatives of the Hladky-Bott torsion T.

If δHT = 0 we say M is of Yang-Mills type,If ∇HT = 0 we say M has horizontally parallel torsion, andIf ∇T = 0 we say M has completely parallel torsion.

LemmaAll H-type foliations are Yang-Mills.


Example: Twistor Spaces

Let (M, g) be a 4n-dimensional (n ≥ 2) quaternionic-Kählermanifold, and fix a quaternionic structure E spanned byI,J ,K ∈ End(TM). Choosing a metric on E so that I,J ,K areorthonormal, we define the twistor space over M to be the unitsphere bundle of E. In this case, we have

Hx ∼= TxM,Vx ∼= CP1.

Here m = dimV = 2 and there is a quaternionic structure inducedby V acting on H.


Further Examples of H-type foliations

rk(V) rk(H) M Torsionany 2k Heisenberg Group C. Parallel1 2k Mixed K-contact Yang-Mills

Sasakian/Hopf fibration C. Parallel2 4k Salamon Twistor Spaces H. Parallel3 4k Mixed 3K-contact Yang-Mills

3-Sasakian/Quaternionic Hopf H. ParallelTorus Bundles over HK C. Parallel

4,5,7 8k Grassmannian Type H. Parallel7 8 Octonionic Hopf H. Parallel


Dimensional Restrictions

For unit Z ∈ V, the maps JZ will induce complex, quaternionic,and octonionic structures on H. As a consequence,

LemmaDenote m = rk(V), n = rk(H). Then

1 m ≤ n − 1,2 m = n − 1 implies n = 2, 4, or 8,3 n = 2k, and furthermore

if m ≥ 2 then n = 4k,if m ≥ 4 then n = 8k.


Clifford Structures

Let (M,H, g) be an H-type foliation with Zi,Zj ∈ V, then

JZiJZj + JZjJZi = −2⟨Zi,Zj⟩IdH

and so we can extend J in the natural way to

J : Cl(V) → End(H)

There is a classification of such Clifford structures over Riemannianmanifolds. (A. Moroianu, U. Semmelmann ‘11 [6])


Parallel Horizontal Clifford StructuresArising from Riemannian or semi-Riemannian foliations withcurvature constancy we have the following notion:

DefinitionLet (M,H, g) be an H-type foliation with horizontally paralleltorsion. Then if there exists a map

Ψ: V × V → Cl2(V)

such that(∇Z1J)Z2 = JΨ(Z1,Z2)

for all Z1,Z2 ∈ Γ(V) then we say that M has a parallel horizontalClifford structure.

This has a strong relationship with the horizontal holonomy of thespace.


Parallel Horizontal Clifford Structures

Moreover, these structures are rigid in that they must be of thefollowing form:

Lemma (Baudoin, Grong, Rizzi, & M. ‘18 [2])Let (M,H, g) be an H-type foliation with parallel horizontalClifford structure. There exists κ ∈ R such that

Ψ(Z1,Z2) = −κ(Z1 · Z2 + ⟨Z1,Z2⟩)

for all Z1,Z2 ∈ Γ(V); moreover the sectional curvature of theleaves associated to V is constant and equal to κ2.


J2 condition

In many model cases, such as the Complex-, Quaternionic-, andOctonionic-Hopf fibrations we have a stronger property:

DefinitionLet (M,H, g) be an H-type foliation. We say that it satisfies theJ2 condition if for every Z1,Z2 ∈ V with ⟨Z1,Z2⟩ = 0 there existsZ3 ∈ V such that

JZ1JZ2 = JZ3

The H-type groups with this property were classified by (M.Cowling, A.H. Dooley, A. Korányi, and F.Ricci ’91 [4])


Metric Connections, Jacobi Equation

For the remainder of the talk, let (M,H, gε)be a sRmc-manifold,equipped with the canonical variation gε,having horizontally parallel torsion,and satisfying the J2 condition.

For any ε > 0 the connection

∇̂εXY = ∇XY + JεXY

will be metric with metric adjoint ∇ε. For a gε-geodesic γ, theJacobi equation in this setting is

∇̂εγ̇∇ε

γ̇W + R̂ε(W, γ̇)γ̇ = 0


The Comparison Principle

Theorem (Baudoin, Grong, Kuwada, & Thalmaier ‘17 [1])Let x, y ∈ M,γ : [0, rε] → M a unit speed gε-geodesic connecting x, y, andW1, · · · ,Wk be a collection of vector fields along γ such that

k∑i=0

∫ rε

0⟨∇̂ε

γ̇∇εγ̇Wi + R̂ε(Wi, γ̇)γ̇,Wi⟩ε ≥ 0

then at y = γ(rε) it holds that

k∑i=0

Hess∇̂ε(rε)(Wi,Wi) ≤

k∑i=0

⟨Wi, ∇̂εγ̇Wi⟩ε

with equality if and only if the Wi are Jacobi fields.


Horizontal Splitting

We introduce an orthogonal splitting of the horizontal bundle.Fixing a vector field Y ∈ H,

H = span(Y)⊕HRiem(Y)⊕HSas(Y)

whereHSas(Y) = {JZY|Z ∈ V}

HRiem(Y) = {X ∈ H|X ⊥ HSas ⊕ span(Y)}

LemmaDenoting n = rk(H),m = rk(V), we will have

dim(HSas) = m, dim(HRiem) = n − m − 1


Comparison Functions

Simliarly to the Riemannian case, we consider the comparisionfunctions

FRiem(r, κ) =

√κ cot(

√κr) if κ > 0

1r if κ = 0√|κ| coth(

√|κ|r) if κ < 0

FSas(r, κ) =

√κ(sin(

√κr)−√

κr cos(√κr))2−2 cos(

√κr)−√

κr sin(√κr) if κ > 04r if κ = 0√κ(

√κr cosh(√κr)−sinh(

√κr))

2−2 cosh(√κr)+√

κr sinh(√κr) if κ < 0

These comparison functions will correspond to the splitting of H.


Hessian Comparisons

Theorem (Baudoin, Grong, Rizzi, & M. ‘19 [3])Let γ : [0, rε] → M be a gε-geodesic. Then

Hess(rε)(γ̇, γ̇) ≤∥γ̇∥2 (1 − ∥γ̇∥2)

rε

If Sec(X ∧ Y) ≥ ρ > 0 for all unit X,Y ∈ HRiem(γ̇), then

Hess(rε)(X,X) ≤ FRiem(rε,K)

If Sec(X ∧ JZX) ≥ ρ > 0 for all unit X ∈ HSas(γ̇), then

Hess(rε)(X,X) ≤ FSas(rε,K)

Where K is a constant depending on ρ, ε, ∥∇V rε∥, and ∥∇Hrε∥.


Horizontal Ricci Curvature

We can define the horizontal Ricci curvature as the trace of theRiemann tensor,

RicH(X,X) =n∑

i=0⟨R∇(Wi,X)X,Wi⟩

= ⟨R∇(Y,X)X,Y⟩+ RicSas(X,X) + RicRiem(X,X)

where the splitting corresponds to the decomposition

H = span(Y)⊕HSas ⊕HRiem


Diameter Estimates

Theorem (Baudoin, Grong, Rizzi, & M. ‘19 [3])Let ρ > 0. Then for unit X ∈ H,

1RicRiem(X,X)

n − m − 1 ≥ ρ =⇒ diam0(M) ≤ π√ρ

2 Sec(X ∧ JZX) ≥ ρ =⇒ diam0(M) ≤ 2π√ρ

3RicSas(X,X)

m ≥ ρ =⇒ diam0(M) ≤ 2π√

3√ρ

and in each case the fundamental group of M must be finite.

The first two of these are sharp, as they are achieved in thecomplex, quaternionic, and octonionic Hopf fibrations.


sub-Laplacian

Similarly to the horizontal Ricci curvature, we can define thesub-Laplacian as the trace of the Hessian. For the distancefunction rε along a geodesic γ with Y = ∇Hrε,

∆Hrε =n∑

i=0Hess(rε)(Wi,Wi)

= Hess(rε)(Y,Y) +m∑

i=0Hess(rε)(JZiY, JZiY) +

n−m−1∑i=0

Hess(rε)(Wi,Wi)

for appropriate bases {Wi} of H and {Zi} of V. This splittingcorresponds again to the decomposition

H = span(Y)⊕HSas ⊕HRiem


Laplacian Comparisons

In each component of the horizontal decomposition we can use theprevious comparisons on the Hessian to obtain

Theorem (Baudoin, Grong, Rizzi, & M. ‘19 [3])Let (M, g,H) be an H-type foliation with parallel horizontal Cliffordstructure and satisfying the J2 condition, and with nonnegativehorizontal Bott curvature. Then there exists a C > 4 such that

∆Hr0 ≤ n − m + 3 + C(m − 1)r0

This is not sharp, but we can recover sharp estimates in eachsubspace.


References I

F. Baudoin, E. Grong, K. Kuwada, and A. Thalmaier.Sub-Laplacian comparison theorems on totally geodesicRiemannian foliations.arXiv e-prints, Jun 2017, 1706.08489.F. Baudoin, E. Grong, G. Molino, and L. Rizzi.H-type Foliations.arXiv e-prints, Dec 2018, 1812.02563.F. Baudoin, E. Grong, G. Molino, and L. Rizzi.Comparison theorems on H-type sub-Riemannian manifolds.arXiv e-prints, Sept 2019, 1909.03532.M. Cowling, A. Dooley, A. Korányi, and F. Ricci.H-type groups and Iwasawa decompositions.Adv. Math., 81(1):1–41, 1991.


References II

R. K. Hladky.Connections and curvature in sub-Riemannian geometry.Houston J. Math., 38(4):1107–1134, 2012.

A. Moroianu and U. Semmelmann.Clifford structures on Riemannian manifolds.Adv. Math., 228(2):940–967, 2011.


Thank you for your attention!

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