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CURVATURE-DIMENSION INEQUALITIES ON

SUB-RIEMANNIAN MANIFOLDS OBTAINED FROM

RIEMANNIAN FOLIATIONS, PART I

ERLEND GRONGANTON THALMAIER

Abstract. We give a generalized curvature-dimension inequality connectingthe geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian mani-folds obtained from Riemannian foliations. We give a geometric interpretationof the invariants involved in the inequality. Using this inequality, we obtaina lower bound for the eigenvalues of the sub-Laplacian. This inequality alsolays the foundation for proving several powerful results in Part II.

1. Introduction

On a given connected manifold M , there is a well established relation betweenelliptic second order differential operators on M and Riemannian geometries onthe same space. More precisely, for any smooth elliptic operator L on M withoutconstant term, there exist a unique Riemannian metric g on M such that for anypair of smooth functions f, g ∈ C∞(M),

(1.1) Γ(f, g) :=1

2(Lfg − fLg − gLf) = 〈gradf, grad g〉g.

Conversely, from any Riemannian metric g, we can construct a second order op-erator satisfying (1.1) in the form of the Laplacian ∆. In addition, the propertiesof L and g are intimately connected. Consider the case when M is complete withrespect to g and write Pt = et/2∆ for the heat semi-group corresponding to 1

2∆.Then the following statements are equivalent for any ρ ∈ R.

(a) For any f ∈ C∞c (M), ρ‖ gradf‖2g ≤ Ricg(grad f, gradf).

(b) For any f ∈ C∞c (M), ‖ gradPtf‖2g ≤ e−ρtPt‖ gradf‖2g.

(c) For any f ∈ C∞c (M), 1−e−ρt

ρ ‖ gradPtf‖2g ≤ Ptf2 − (Ptf)

2.

Here, C∞c (M) is the space of smooth functions on M with compact support and

Ricg is the Ricci curvature tensor of g. This equivalence gives us a way of under-standing Ricci curvature in terms of growth of the gradient of a solution to the heatequation. With appropriate modifications of the Ricci curvature, the same state-ment holds for a general elliptic operator L satisfying (1.1), giving us a geometrictool to study the heat flow of elliptic operators. See e.g. [27] and references thereinfor the full statement.

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http://de.arxiv.org/abs/1408.6873v1

2 E. GRONG, A. THALMAIER

Let us now consider the case when L is not elliptic, but is rather locally givenon a form

(1.2) L =

n∑

i=1

X2i + first order terms.

Here, X1, . . . Xn are linearly independent vector fields and n is less than the dimen-sion of M . In this case, there is still a geometry we can associate with L by con-sidering the subbundle spanned by X1, . . . , Xn furnished with a metric tensor thatmakes these vector fields orthogonal. Such a geometry is called sub-Riemanniangeometry. A sub-Riemannian manifold is a connected manifold M with a positivedefinite metric tensor h defined only on a subbundle H of the tangent bundle TM .If we assume that sections of H and their iterated Lie brackets span the entiretangent bundle, we obtain a metric space (M, dcc), where the distance between twopoints is defined by taking the infimum of the lengths of all curves tangent to Hthat connect these points.

In recent years, understanding how to define curvature in sub-Riemannian ge-ometry has become an important question. One approach has been to introducecurvature by studying invariants of the flow of normal geodesics associated to thesub-Riemannian structure, see e.g. [28, 19, 2, 5]. The other approach explores theinteraction of the sub-Riemannian gradient and second order operators on the form(1.2). We will follow the latter approach.

The equivalence of statements (a), (b) and (c) mentioned previously is rooted inthe curvature-dimension inequality for Riemannian manifolds. In the notation ofBakry and Emery [3] this inequality is written as

Γ2(f) ≥1

n(Lf)2 + ρΓ(f), f ∈ C∞(M),

Here, n = dimM,L = ∆, ρ is a lower bound for the Ricci curvature, and for anypair of functions f, g ∈ C∞(M)

Γ(f, g) =1

2(L(fg)− fLg − gLf) = 〈grad f, grad g〉g, Γ(f) = Γ(f, f),(1.3)

Γ2(f, g) =1

2(LΓ(f, g)− Γ(f, Lg)− Γ(Lf, g)) , Γ2(f) = Γ2(f, f).(1.4)

Even in simple cases, this inequality fails in the sub-Riemannian setting. Thefollowing generalization has been suggested by F. Baudoin and N. Garofalo in [6].

Let h be a sub-Riemannian metric defined on a subbundle H of TM . Let L beany second order operator as in (1.2). We remark that unlike the Laplace operatoron a Riemannian manifold, the operator L is not uniquely determined by h unlesswe add some additional structure such as a chosen preferred volume form vol onM .Define Γ and Γ2 as in (1.3) and (1.4). For any positive semi-definite section v∗ ofSym2 TM , define Γv

∗

(f, g) = v∗(df, dg) and Γv∗

(f) = Γv∗

(f, f). Let Γv∗

2 (f) bedefined analogous to the definition Γ2(f) in (1.4). Then L is said to satisfy ageneralized curvature-dimension inequality if we can choose v∗ such that for everyℓ > 0,

(1.5) Γ2(f) + ℓΓv∗

2 (f) ≥ 1

n(Lf)2 +

(ρ1 − ℓ−1

)Γ(f) + ρ2Γ

v∗

(f),

for some 1 ≤ n ≤ ∞, ρ1 ∈ R and ρ2 > 0. Using this inequality, the authors areable to prove several results, such as gradient bounds, Li-Yau type inequality and

CURVATURE-DIMENSION INEQUALITIES ON SR-MANIFOLDS, PART I 3

a sub-Riemannian version of the Bonnet-Myers theorem. See also further resultsbased of the same formalism in [7, 8, 9], see also [15].

So far, the examples of sub-Riemannian manifolds satisfying (1.5) all have acomplement to H spanned by the sub-Riemannian analogue of Killing vector fields.We want to show that a further generalization of (1.5) holds for a larger class ofsub-Riemannian manifolds. We also want to give an interpretation for the con-stants involved in the curvature-dimension inequality. Results following from thisinequality are important for sub-Riemannian geometry, understanding solutionsof the heat equation of operators L described locally as in (1.2), along with thestochastic processes which have these operators as their infinitesimal generators.

In order to motivate our approach, let us first consider the following example.Let π : M → B be a submersion between two connected manifolds and let qg be aRiemannian metric on B. Let V = kerπ∗ be the vertical bundle and let H be anEhresmann connection on π, i.e. a subbundle such that TM = H ⊕ V . We thendefine a sub-Riemannian metric h on M by pulling back the Riemannian metric,i.e. h = π∗qg|H. In this case, we have two notions of curvature that could be expectedto play a role for the inequality of type (1.5), namely the Ricci curvature of B andthe curvature of the Ehresmann connection H (see Section 3.1 for definition). Afterall, our sub-Riemannian structure is uniquely determined by a metric on B and achoice of Ehresmann connection. For this reason, examples of this type shouldbe helpful in providing a geometric understanding of curvature in sub-Riemanniangeometry. However, we have to deal with the following two challenges.

(i) Even though the sub-Riemannian geometry onM can be considered as “lifted”from B the same cannot be said for our operator L. That is, if L is of the

type (1.2), q∆ is the Laplacian on B and f ∈ C∞(B) is a smooth function on

B, then L(f π) does not coincide with (q∆f) π in general.(ii) The same sub-Riemannian structure on M can sometimes be considered as

lifted from two different Riemannian manifolds (see Section 4.5 for an exam-ple).

Our approach to overcome these challenges will be the following.In Section 2, we introduce the basics of sub-Riemannian manifolds and sub-

Laplacians. We overcome the challenges of (i) and (ii) by introducing a uniqueway of choosing L relative to a complemental subbundle of H rather than a vol-ume form. This will have exactly the desired “lifting property”. We discuss thediffusions of such operators in terms of stochastic development. In Section 3 we in-troduce a preferable choice of complement, which we call metric-preserving comple-ments. Roughly speaking, such complements correspond to Riemannian foliations.While such a complement may not always exist, all sub-Riemannian manifolds dis-cussed so far have such a complement. We give geometric conditions for whena sub-Riemannian manifold with a metric-preserving complement satisfies a gen-eralization of the curvature-dimension inequality (1.5). From this inequality, weimmediately get a result on the spectral gap of L found in Section 4, along withexamples.

In Part II we will look at further consequences of the curvature-dimension in-equality in Theorem 3.5. A short summary of these results are given Section 5

1.1. Notations and conventions. Unless otherwise stated, all manifolds are con-nected. If E →M is a vector bundle over a manifoldM , its space of smooth sections


is written Γ(E). If s ∈ Γ(E) is a section, we generally prefer to write s|m ratherthan s(m) for its value in m ∈ M . By a metric tensor s on E , we mean a smoothsection of Sym2 E∗ which is positive definite or at least positive semi-definite. Forevery such metric tensor, we write ‖e‖s =

√s(e, e) for any e ∈ E even if s is only

positive semi-definite. All metric tensors are denoted by bold, lower case Latinletters (e.g. h,g, . . . ). We will only use the term Riemannian metric for a positivedefinite metric tensor on the tangent bundle. If g is a Riemannian metric, we will

use g∗, ∧k g∗, . . . for the metric tensors induced on T ∗M,∧k

T ∗M, . . . .If α is a form on a manifold M , its contraction or interior product by a vector

field X will be denoted by either ιXα or α(X, ·). We use LX for the Lie derivativewith respect to X .

Stochastic processes are denoted by bold capital Latin letters (e.g. X,Y, . . . ).Stopping times are denoted by small bold Greek letters (τ ,σ, . . . ).

2. Sub-Riemannian manifolds and sub-Laplacians

2.1. Definition of a sub-Riemannian manifolds. A sub-Riemannian manifold

can be considered as a triple (M,H,h) where M is a connected manifold, H is asubbundle of TM and h is a positive definite metric tensor defined only on thesubbundle H. The pair (H,h) is called a sub-Riemannian structure on M . Anysub-Riemannian structure induces a vector bundle morphism

♯h∗

: T ∗M → TM,

determined by the properties ♯h∗

(T ∗M) = H and p(v) = h(v, ♯h∗

p) for any p ∈T ∗M and v ∈ H. The kernel of ♯h

∗

is the subbundle Ann(H) ⊆ T ∗M of allelements of T ∗M that vanish on H. We can define a co-metric h∗ on T ∗M by

h∗(p1, p2) = p1(♯h∗

p2), p1, p2 ∈ T ∗mM, m ∈M,

which obviously degenerates along Ann(H). A sub-Riemannian manifold can there-fore equivalently be considered as a pair (M,h∗) where M is a connected manifoldand h∗ a co-metric degenerating along a subbundle of T ∗M . We will use both ofthese point of views throughout our paper, referring to the sub-Riemannian struc-ture (H,h) and h∗ interchangeably.

We will call any absolutely continuous curve γ in M horizontal if γ(t) ∈ Hγ(t)

for almost all t. We define the Carnot-Caratheodory distance dcc on M as

dcc(m0,m1) = infγ

∫ 1

0

h(γ, γ)1/2 dt : γ(0) = m0, γ(1) = m1, γ horizontal

.

This distance is finite for any pair of points if they can be connected by at least onehorizontal curve. A sufficient condition for the latter to hold is that H is bracket-generating [11, 22]. A subbundle H is called bracket-generating if its sections andtheir iterated brackets span the entire tangent bundle. The same property alsoguarantees that the metric topology induced by dcc coincides with the manifoldtopology on M , however, the Hausdorff dimension of dcc will in general be greaterthan the topological dimension (see e.g. [21, Th. 2.3, Th. 2.17 ]).

From now on, the rank of H is n ≥ 2 while the manifold M is assumed to havedimension n + ν. We will refer to H as the horizontal bundle and its vectors andsections as horizontal, and we refer to both (H,h) and h∗ as a sub-Riemannianstructure on M .


2.2. Second order operators associated to h∗. For any manifold M , let T 2Mdenote the second order tangent bundle. Sections L ∈ Γ(T 2M) of this bundle canlocally be expressed as

(2.1) L =

n+ν∑

i,j=1

Lij∂

∂xi∂xj+

n+ν∑

j=1

Lj∂

∂xj,

relative to some local coordinate system (x1, . . . , xn+ν) and functions Lij = Ljiand Lj . We can consider TM as a subbundle of T 2M . Denote its inclusion by inc.This gives us a short exact sequence

0 → TMinc−→ T 2M

q−→ Sym2 TM → 0,

where q(L) =: qL is the symmetric, bilinear tensor on T ∗M , defined by the property

(2.2) qL(df, dg) =1

2(L(fg)− fLg − gLf) , for any f, g ∈ C∞(M).

In local coordinates, we can write qL(df, dg) =∑n+ν

i,j=1 Lij∂f∂xi

∂g∂xj

relative to the

representation of L in (2.1).Let h∗ be the co-metric corresponding to a sub-Riemannian structure (H,h).

Then any operator L satisfying qL = h∗ can locally be written as

L =

n∑

i=1

X2i +X0

where X0 is a vector field and X1, . . . , Xn is a local orthonormal basis of H. FromHormander’s celebrated result [16], we know that any such operator is hypoellipticwhen H is bracket-generating. We consider two examples where a choice of extrastructure on (M,H,h) gives a differential operator of this type.

Let vol be a volume form on M . We define the sub-Laplacian relative to vol asthe operator

∆H = div ♯h∗

df,

where divX is defined by LX vol = (divX) vol. Any such operator satisfies∫

M

g∆Hf dvol =

∫

M

f∆Hg dvol

for any pair of functions f, g ∈ C∞c (M) of compact support. Since L is also hy-

poelliptic, it has a smooth, symmetric heat kernel with respect to vol . This is themost common way of defining the sub-Laplacian.

We like to introduce an alternative notion of sub-Laplacian. Rather than choos-ing a volume, we will choose a complement V to H, i.e. a subbundle V such thatTM = H ⊕ V . This choice of complement gives us projections prH and prV torespectively H and V . A Riemannian metric g on M is said to tame h if g |H = h.Consider any Riemannian metric g that tames h and makes V the orthogonal com-plement of H. Let ∇ be the Levi-Civita connection of g. It is simple to verify thatfor any pair of horizontal vector fields X,Y ∈ Γ(H), prH ∇XY is independent ofg |V . This fact allows us to define a second order operator ∆′

H which we call thesub-Laplacian with respect to V . There are several ways to introduce this opera-

tor. We have chosen to define it by using a connection ∇ which will be helpful for


us later. Corresponding to the Riemannian metric g and the orthogonal splittingTM = H⊕⊥ V , we introduce the connection

∇XY := prH∇prHX prH Y + prV ∇pr

VX prV Y(2.3)

+ prH[prV X, prH Y ] + prV [prHX, prV Y ].

Definition 2.1. Let V be a complement of H corresponding to the projection prH,

i.e. V = ker prH. Then the sub-Laplacian with respect to V is the operator

∆′Hf := tr pr∗H ∇2· , ·f, f ∈ C∞(M),

where ∇2X,Y = ∇X∇Y − ∇∇XY

is the Hessian of ∇.

We remark that the definition only depends on the value of ∇XY when both Xand Y are take values in H. This is illustrated by the fact that locally

(2.4) ∆′Hf =

n∑

i=1

X2i f +

n∑

i,j=1

h(prH ∇XiXj , Xi)Xjf.

whereX1, . . . , Xn is a local orthonormal basis ofH. The operator ∆′H is hypoelliptic

and will have a smooth heat kernel with respect to any volume form. We also notethat two different choices of complement may have the same sub-Laplacian, seeSection 4.5.

In what follows, whenever we have a chosen complement V , we will refer to it asthe vertical bundle and its vectors and vector fields as vertical.

Remark 2.2. The horizontal bundle H is called regular if there exist a flag of sub-bundles

H = H1 ⊆ H2 ⊆ H3 ⊆ · · · ,such that

Hk+1 = spanY |m, [X,Y ]|m : Y ∈ Γ(Hk), X ∈ Γ(H), m ∈M.The smallest integer r such that Hr = TM is called the step of H. If (H,h) is asub-Riemannian structure onM with H regular, then there exist a canonical choiceof volume form on M called Popp’s measure. For construction, see Section 4.2 orsee [1] for a more detailed presentation.

2.3. Lifting property of the sub-Laplacian defined relative to a comple-

ment. Let π : M → B be a surjective submersion between connected manifoldsM and B. The vertical bundle of π is the subbundle V := kerπ∗ of TM . An

Ehresmann connection on π is a splitting h of the short exact sequence

0 −→ V = kerπ∗ −→ TM π∗

// π∗TBh

ss −→ 0.

This map h is uniquely determined by its image H = image h, which is a subbundleof TM satisfying TM = H ⊕ V , so we will often refer to H as the Ehresmannconnection as well. The image of an element (m, v) ∈ π∗TB under h is called thehorizontal lift of v to m, and denoted hmv. Similarly, for any vector field X on B,we have a vector field hX on M defined by m 7→ hmX|π(m).

We can extend the notion of horizontal lifts to second order vectors and differen-tial operators. If B and M are two manifolds, then a linear map A : T 2

bB → T 2mM


is called a Schwartz morphism if A(TbM) ⊆ TmM and the following diagram com-mutes

0 // TmMinc

// T 2mM

q// Sym2 TmM // 0

0 // TbBinc

//

A|TbB

OO

T 2bB

q//

A

OO

Sym2 TbB //

A|TbB⊗A|TbB

OO

0

with q defined as in (2.2). We remark that any linear map A : T 2bB → T 2

mM isa Schwartz morphism if and only if A = f∗|T 2

bB for some map f : B → M with

f(b) = m (see e.g. [13, p. 80]). Let π : M → B be a surjective submersion. A2-connection on π is then a splitting hS of the short exact sequence

0 −→ kerπ∗ −→ T 2M π∗

// π∗T 2BhS

rr −→ 0.

such that hS is a Schwartz morphism on any point.For any choice of Ehresmann connection h on π, we can construct a corresponding

2-connection hS uniquely determined by the following two requirements (see e.g.[20, pp 82–83]).

• hS |TM = h,• hSX, Y = hX, hY where X, Y = 1

2 (XY + Y X) is the skew-commutator

and X, Y ∈ Γ(TB). Equivalently, hS(XY ) = hXhY − 12 prV [hX, hY ].

Using this 2-connection, we can define horizontal lifts of second order operatorson B. We then have the following way to interpret the sub-Laplacian with respectto a complement.

Proposition 2.3. Let qg be a Riemannian metric on B with Laplacian q∆. Relative

to an Ehresmann connection H on π, define a sub-Riemannian structure (H,h) byh = π∗ g |H. Then hS q∆ = ∆′

H where ∆′H is the sub-Laplacian of V = kerπ∗.

We remark in particular that for any f ∈ C∞(B), we have ∆′H(f π) = (q∆f)π.

Note that if π : (M,g) → (B, qg) is a submersion between two Riemannian mani-folds such that g |H = π∗qg|H where H = (kerπ∗)⊥, then π is called a Riemannian

submersion. The sub-Riemannian manifolds of Proposition 2.3 can hence be con-sidered as the result of restricting the metric on the top space in a Riemanniansubmersion to its horizontal subbundle.

Proof of Proposition 2.3. Let g be a Riemannian metric on M satisfying g |H = h

and H⊥ = V with respect to g. Let X1, . . . , Xn be any local orthonormal basis ofTB. Then the Laplacian can be written as

q∆ =

n∑

i=1

X2i +

n∑

i,j=1

qg(q∇XiXj , Xi)Xj

where q∇ is the Levi-Civita connection of qg. However, since g(hXi, hXj) = qg(Xi, Xj)

and since prH[hXi, hXj ] = h[Xi, Xj ], we obtain

qg(q∇XiXj , Xi) = g(∇hXi

hXj, hXi).

The result follows from (2.4) and the fact that hX1, . . . , hXn is a local orthonormalbasis of H.


Since the proof of Proposition 2.3 is purely local, it also holds on Riemannianfoliations. To be more specific, a subbundle V of TM is integrable if [V,W ] takesits values in V whenever V,W ∈ Γ(V) are vertical vector fields. From the Frobeniustheorem, we know that there exists a foliation F of M consisting of immersedsubmanifolds of dimension ν = rankV such that each leaf is tangent to V .

A Riemannian metric g onM with a foliation induced by V , is called bundle-like

if V⊥ = H and for any X ∈ Γ(H) and V ∈ Γ(V), we have (LV g)(X,X) = 0. In-tuitively, one can think of a bundle-like metric g as a metric where g |H does “notchange” in vertical directions. A foliation F onM is called Riemannian if a bundle-like metric exists. Such a manifold locally has the structure of a Riemannian sub-mersion, that is, any point has a neighborhood U such that π : U → B := U/(F|U)can be considered as a smooth submersion of manifolds, H is an Ehresmann on πand B can be given a Riemannian metric qg such that pr∗H g = π∗qg, see [23]. If wedefine a sub-Riemannian structure (H,h) on M with h = g |H, then restricted to

each sufficiently small neighborhood U , the sub-Laplacian ∆′H of V is equal to hS q∆

were q∆ is the Laplacian on U/(F|U).

Remark 2.4. By modifying the proof of Proposition 2.3 slightly, we can also getthe following stronger statement: Let (B,H1,h1) be a sub-Riemannian manifoldand let π : M → B be a submersion with an Ehresmann connection E . Define asubbundle H2 on M as horizontal lift of all vectors in H1 with respect to E and leth2 = π∗h1|H2

be the lifted metric. Let V1 be a choice of complement of H1 anddefine V2 as the direct sum of the horizontal lift of V1 and kerπ∗. Then, if ∆′

Hjis

the sub-Laplacian with respect to Vj, j = 1, 2, we have ∆′H2

= hS∆′H1.

2.4. Comparison between the sub-Laplacian of a complement and a vol-

ume form. Let (M,H,h) be a sub-Riemannian manifold and let g be a Riemann-ian metric taming h. Let ∆H be the sub-Laplacian defined with respect to thevolume form of g and let ∆′

H be defined relative to H⊥ = V . We introduce a vectorfield N by formula

N = ∆′H −∆H.

We give two descriptions of this vector field, which can easily be verified from thedefinition.

First, the vector fieldN ∈ Γ(TM) is horizontal and can be defined by the relation

(2.5) g(X,N) = −1

2tr(Lpr

HX g)(prV · , prV ·).

Secondly, assume that H and V are orientable (we can always assume this locally,which will be sufficient for us). Let χH ∈ Γ(

∧nH) and χV ∈ Γ(∧ν V) be the unique

positively oriented multivectors of unit length. Then

N = ♯h∗

ιχVdιχH

vol .

From the latter realization, it follows that if N = (−1)ν♯h∗

dφ for φ ∈ C∞(M), then∆′

H is symmetric with respect to volume form eφ vol .

Remark 2.5. Let (M,g) be a Riemannian manifold with a foliation given by anintegrable subbundle V . Assume that g is bundle-like relative to V . Let H be theorthogonal complement of V with respect to g and let h = g |H. Write vol for thevolume form of g. Let ∆H and ∆′

H be the sub-Laplacian of (M,H,h) relative torespectively vol and V . Then N = ∆′

H −∆H is the mean curvature vector field of


the leaves of the foliations by (2.5). Hence, the operators ∆H and ∆′H coincide in

this case if and only if the leafs of the foliation are minimal submanifolds.

2.5. Diffusion of ∆′H. Let L be any section of T 2M with qL being positive semi-

definite and let m ∈M be any point. Then, by [17, Th 1.3.4, Th 1.3.6] there existL-diffusion Xt = Xt(m) satisfying X0 = m, unique in law, defined up to someexplosion time τ = τ (m). An L-diffusion Xt is an M -valued semimartingale up tosome stopping time τ defined on some filtered probability space (Ω,F· ,P), suchthat for any f ∈ C∞(M),

Mft := f(Xt)− f(X0)−

∫ t

0

Lf(Xs)ds, 0 ≤ t < τ ,

is a local martingale up to τ . We will always assume that τ is maximal, so that τis the explosion time, i.e. τ <∞ ⊆ limt↑τ Xt = ∞ almost surely.

If ∆′H is the sub-Laplacian defined with respect to a choice of complement V ,

let g be a Riemannian metric such that H⊥ = V and g |H = h. Define ∇ on as

in (2.3). To simplify our presentation, we will assume that ∇g = 0. See Remark 2.6for the general case. For a given point m ∈M , let γ(t) be any smooth curve in Mwith γ(0) = m. Let φ1, . . . , φn and ψ1, . . . , ψν be orthonormal bases for respectivelyH|m and V|m. Parallel transport of such bases remain orthonormal bases from our

assumption ∇g = 0.Define O(n) → O(H) →M as the bundle of orthonormal frames ofH, and define

O(ν) → O(V) →M similarly. Let

O(n)×O(ν) → O(H)⊙O(V) π→M,

denote the product bundle. We can then define an Ehresmann connection E∇ on πsuch that a curve (φ(t), ψ(t)) = (φ1(t), . . . , φn(t), ψ1(t), . . . , ψν(t)) in O(H)⊙O(V)is tangent to E∇ if and only if each φj(t), 1 ≤ j ≤ n and ψs(t), 1 ≤ s ≤ ν is parallelalong γ(t) = π(φ(t), ψ(t)).

Define vector fields X1, . . . , Xn on O(H)⊙O(V) by Xj|φ,ψ = hφ,ψφj , where the

horizontal lift is with respect to E∇. For any m ∈M and (φ, ψ) ∈ O(H)⊕O(V)|m,consider the solution Ft of the Stratonovich SDE up to explosion time τ ,

dFt =n∑

j=1

Xj |Ft dBj

t , F0 = (φ, ψ),

where B = (B1, . . . ,Bn) is a Brownian motion in Rn with B0 = 0. It is then

simple to verify that F is a 12h

S∆′H-diffusion since hS∆′

H =∑ni=1 X

2j if we consider

the 2-connection hS induced by the Ehresmann connection E∇. This shows thatXt = π(Ft) is an 1

2∆′H-diffusion on M with X0 = m. Note that τ will be the

explosion time of X as well by [24].

Remark 2.6. If ∇g 6= 0, we can instead use the connection

(2.6) ˚∇XY := prH ∇X prH Y + prV ∇X prV Y, X, Y ∈ Γ(TM).

It clearly satisfies˚∇g = 0, preserve the horizontal and vertical bundle under paralleltransport and has ∇XY =˚∇XY for any pair horizontal vector fields X,Y ∈ Γ(H).

The definition of ∆H in Definition 2.1 hence remains the same if we replace ∇ with˚∇. The reason we will prefer to use ∇ is the property given in Lemma 3.2(a) which

fails for ˚∇.


Remark 2.7. Instead of using the lift to O(H) ⊙ O(V), we could have considered

the full frame bundle O(TM) and development with respect to ∇ (or ˚∇), see e.g.[17, Section 2.3]. The diffusion of 1

2∆′H then has the Brownian motion in an n-

dimensional subspace of Rn+ν as its anti-development, where the subspace dependson the choice of initial frame.

Remark 2.8. If ∆′H is also symmetric with respect to some volume form vol, the

the following observation made in [9, Th 4.4] guarantees us that the diffusion X hasinfinite lifetime, i.e. τ = ∞ a.s.. Let g be any Riemannian metric on M tamingh. Assume that g is complete with (Riemannian) Ricci curvature bounded frombelow. Note that if dg is the metric of g and dcc is the Carnot-Caratheodory of h,then dcc(m,m1) ≥ dg(m,m1) for any (m,m1) ∈ M ×M . Hence, Br(m) ⊆ Bg

r (m)where Br(m) and Bg

r (m) are the balls of respectively dcc and dg, centered at mwith radius r. Hence, by the Riemannian volume comparison theorem, we have

vol(Br(m)) ≤ vol(Bgr (m)) ≤ C1e

C2r.

for some constants C1, C2. Hence,∫∞0

rlog vol(Br(m))dr = ∞ and so [26, Th 3] tells

us that 12∆

′H-diffusions X starting at a point m ∈M has infinite lifetime.

3. Riemannian foliations and curvature-dimension inequality

3.1. Riemannian foliations and the geometry of ∇. Let (M,H,h) be a sub-Riemannian manifold with a complement V . Let ∆′

H be the sub-Laplacian relativeto V . In order to introduce a curvature-dimension inequality for ∆′

H, we will needto choose a Riemannian metric onM which tame h and makes H and V orthogonal.Choose a metric tensor v on V to obtain a Riemannian metric g = pr∗H h+ pr∗V v.

We make the following assumptions on V . We want to consider the specific casewhen V is integrable and satisfy

(3.1) LV (pr∗H h) = 0 for any V ∈ Γ(V).Then g = pr∗H h+ pr∗V v is bundle-like for any choice of v, giving us a Riemannianfoliation as defined in Section 2.3. Since this property is really independent of g |V ,we introduce the following definition.

Definition 3.1. An integrable subbundle V is called a metric-preserving comple-

ment to the sub-Riemannian manifold (M,H,h) if TM = H⊕ V and (3.1) holds.

In the special case when the foliation F of V gives us a submersion π : M →B = M/F with H as an Ehresmann connection on π, the curvature of H is a

vector-valued two-form R ∈ Γ(∧2

T ∗M ⊗ TM) defined by

(3.2) R(X,Y ) := prV [prHX, prH Y ], X, Y ∈ Γ(TM).

This curvature measures how farH is from being a flat connection (i.e. an integrablesubbundle). We will call R given by formula (3.2) the curvature of H even when Vdoes not give us a submersion globally.

Define ∇ relative to g as in (2.3). The following properties are simple to verify.

Lemma 3.2. Let g be a Riemannian metric and let V be an integrable subbundle

of TM with orthogonal complement H. Define ∇ relative to g and the splitting

TM = H⊕ V. Write h and v for the restriction of g to respectively H and V.


(a) Let Y ∈ Γ(TM) is an arbitrary vector field. Then, if X ∈ Γ(H) is horizontal,

both ∇XY and ∇YX only depends on h and the splitting TM = H⊕V. It doesnot depend on v. Similarly, if V is vertical, then ∇V Y and ∇Y V is independent

of h.

(b) The torsion of ∇ is given as T ∇(X,Y ) = −R(X,Y ).(c) V is a metric-preserving complement of (M,H,h) if and only if

(∇X g)(Y, Y ) = (LprHX g)(prV Y, prV Y ).

Equivalently, V is metric-preserving if and only if ∇h∗ = 0.(d) If V is metric-preserving, then

(∇X g)(Y, Z) = −2 g(X, II(prV Y, prV Z)),

where II is the second fundamental form of the foliation of V.

Recall that when V is metric-preserving, g is bundle-like. We write down the

basic properties of the curvature R∇ of ∇ when V is metric-preserving.

Lemma 3.3. Let Y, Z ∈ Γ(TM) be arbitrary vector fields and let X ∈ Γ(H) be a

horizontal vector field. Then

(a) g(R∇(Y, Z)X,X) = 0.

(b) g(R∇(X,Y )Z,X)− g(R∇(X,Z)Y,X) = 0, which in particular means that

g(R∇(X, prV Z)Y,X) = 0.

Proof. The statement in (a) holds since (∇ g)(prH ·, prH ·) = 0. For the identityin (b), we recall the first Bianchy identity for connections with torsion,

R∇(X,Y )Z = − T ∇(X,T ∇(Y, Z))+ (∇XT∇)(Y, Z),

where denotes the cyclic sum. This means that

g( R∇(X,Y )Z,X) = g(R∇(X,Y )Z,X)− g(R∇(X,Z)Y,X)

= g(− T ∇(X,T ∇(Y, Z))+ (∇XT

∇)(Y, Z), X)= 0.

We will use the fact that we have a clear idea of what Ricci curvature is on aRiemannian manifold, to introduce a corresponding tensor on a sub-Riemannianmanifold with a metric-preserving complement V .

Proposition 3.4. Introduce a tensor RicH ∈ Γ(T ∗M⊗2) by

RicH(Y, Z) = trR∇(prH · , Z)Y.

Then

(a) RicH is symmetric and V ⊆ kerRicH.

(b) RicH is independent of choice of metric v on V.(c) Let F be the foliation induced by V. Let U be any neighborhood of M such

that the quotient map π : U → B := U/(F|U) is a smooth submersions of

manifolds. Let Ric be the Ricci curvature on B with respect to the induced

Riemannian structure. Then RicH |U = π∗Ric.


Proof. From the fact that ∇ preserve both the vertical and horizontal bundle andfrom Lemma 3.3(b), we know that RicH = pr∗H RicH. We know that it is symmetricby Lemma 3.3 (b). The statement (b) can be verified using the definition of theLevi-Civita connection.

To prove (c), let X1, . . . , Xn be any local orthonormal basis on B. Note that

[hXi, hXj] = h[Xi, Xj ] + R(hXi, hXj). Also, for any V ∈ Γ(V), ∇V hXi =

prH[V, hXi] = 0 since hXi and V are π-related to respectively Xi and the zero-section of TB. Then for any k, l,

n∑

i=1

g(R∇(hXi, hXl)hXk, hXi)

=

n∑

i=1

g([

∇hXi, ∇hXl

]hXk − ∇h[Xi,Xl]

hXk − ∇R(hXi,hXl)hXk, hXi

)

=n∑

i=1

qg([

q∇Xi, q∇Xl

]Xk − q∇[Xi,Xl]

Xk, Xi

)= Ric(Xk, Xl).

It follows that RicH(hXk, hXl) = Ric(π∗hXk, π∗hXl), and hence the same will holdfor any pair of vector fields Y, Z.

3.2. A generalized curvature-dimension inequality. For any symmetric bi-linear tensor s∗ ∈ Γ(Sym2 TM), we associate a map Γs

∗

: C∞(M) × C∞(M) →C∞(M) by

Γs∗

: C∞(M)× C∞(M) → C∞(M)(f, g) 7→ s∗(df, dg)

.

By the Leibniz identity, it satisfies Γs∗

(f, gh) = gΓs∗

(f, h) + hΓs∗

(f, g). Relative tosome second order operator L ∈ Γ(T 2M), we define

Γs∗

2 (f, g) :=1

2

(LΓs

∗

(f, g)− Γs∗

(Lf, g)− Γs∗

(f, Lg)).

To simplify notation, we will write Γs∗

(f, f) = Γs∗

(f) and Γs∗

2 (f, f) = Γs∗

2 (f).Let h∗ = qL where q is defined as in (2.2). Assume that h∗ is positive semi-

definite and let v∗ ∈ Γ(Sym2 TM) be another chosen positive semi-definite section.Then L is said to satisfy a generalized curvature-dimension inequality with param-eters n, ρ1, ρ2,0 and ρ2,1 if

Γh∗

2 (f) + ℓΓv∗

2 (f) ≥ 1

n(Lf)2 +

(ρ1 −

1

ℓ

)Γh

∗

(f) + (ρ2,0 + ρ2,1ℓ)Γv∗

(f),(CD*)

for any ℓ > 0. Note that we include the possibility of n = ∞. Any such inequalityimplies Γv

∗

2 (f) ≥ ρ2,1Γv∗

(f) by dividing both sides with ℓ and letting it go toinfinity.

Let (M,H,h) be a sub-Riemannian manifolds with an integrable complement Vthat is also metric-preserving. Choose a metric v on V , and define ∇ with respectto the corresponding Riemannian metric g. Let v∗ be the co-metric correspondingto v. Using the properties of ∇, we are ready to present our generalized curvature-dimension inequality. We will make the following assumptions on (M,H,h).


(i) Let R be the curvature of H relative to the complement V . Assume that thelength of R is bounded on M , i.e

supM

‖R‖∧2 g∗ ⊗g = MR <∞.

From now on, we normalize the vertical metric v by requiring MR = 1.(ii) Let RicH be defined as in Proposition 3.4. Assume that RicH has a lower

bound ρRicH , i.e. for every v ∈ TM , we have

RicH(v, v) ≥ ρRicH‖ prH v‖2h.

(iii) Assume that the length of the tensor ∇· g∗ (= ∇v∗) is bounded. Write

M∇v∗ = supM

∥∥∥∇·v∗(· , ·)∥∥∥g∗ ⊗ Sym2 g∗

.

(iv) Define

(∆′Hv∗)(p, p) = tr(∇2

prH· , pr

H·v∗)(p, p).

and assume that for any p ∈ T ∗M , we have (∆′Hv∗)(p, p) ≥ ρ∆′

Hv∗‖p‖2v∗

globally on M for some constant ρ∆′

Hv∗ .

(v) Finally, introduce RicHV as

RicHV(Y, Z) =1

2tr(g(Y, (∇·R)(· , Z)) + g(Y, (∇·R)(· , Z))

),

Assume then that for any Y ∈ Γ(TM), then globally on M

RicHV(Y, Y ) ≥ −2MRicHV‖ prV Y ‖‖ prH Y ‖,

for some number MRicHV.

Note that MR,M∇v∗ and MRicHVare always non-negative, while this is not nec-

essarily true for ρRicH and ρ∆′

Hv∗ . We will define one more constant, which will

always exist. For any α ∈ Γ(T ∗M), define mR as the maximal number satisfying

‖α(R(· , ·))‖∧2h∗ ≥ mR‖α‖v∗ .

Note that if rankV = ν, then√νmR ≤ MR = 1 and that it can only be nonzero if

H is step 2 regular as defined in Remark 2.2.With these assumptions in place, we have the following version of a generalized

curvature-dimension inequality.

Theorem 3.5. Define Γs∗

2 with respect to L = ∆′H. Then L satisfies (CD*) with

n = rankH and

(3.3)

ρ1 = ρRicH − c−1,

ρ2,0 =1

2m2R − c(MRicHV

+ M∇v∗)2,

ρ2,1 =1

2ρ∆′

Hv∗ − M

2∇v∗

,

for any positive c > 0.

Note that we include the possibility c = ∞ when MRicHV= M∇v∗ = 0. The

proof is found in Section 3.5.We can give the following interpretation of the different terms in the inequality.


(i) MR and mR measures how well v can be controlled by the curvature Rof H. To be more precise, for any m ∈ M , p ∈ T ∗

mM , define C1(m) andC2(m) such that

C1(m)‖p‖v∗ ≤ ‖p R‖∧2h∗ ≤ C2(m)‖p‖v∗ .

Then MR ≥ supM C2(m), while mR = infM C1(m).(ii) RicH is a generalization of “the Ricci curvature downstairs” on sub-Riemannian

structures on submersions by Proposition 3.4(b).(iii,iv) Both M∇v∗ and ρ∆′

Hv∗ measure how v changes in horizontal directions. In

particular, ∇v∗ is the second fundamental form by Lemma 3.2.(v) RicHV measures how “optimal” our subbundle H is with respect to our cho-

sen complement V in the sense that on invariant sub-Riemannian structureson principal bundles, RicHV measures how far H is from being a Yang-Millsconnection, see Example 4.3. We will see how RicHV can be interpreted ina similar way in the general case in Appendix A.4, Part II.

For further geometric interpretation, see Part II, Section 5.2.

Remark 3.6. In the proof of Theorem 3.5, we prove a curvature-dimension inequalitywithout normalizing MR in (3.9). The reason why we are free to normalize v such

that MR = 1 is the following. Since ∇XY is independent of v when either X orY are horizontal, the bounds introduced in (i)-(v) behave well under scaling in thesense that for any ε > 0, if define the bounds relative to v2 = 1

εv rather than v,

we will get the same inequality back for Γh∗

2 (f) + ℓεΓ

v∗

2

2 (f) = Γh∗

2 (f) + ℓΓv∗

2 (f).

3.3. The special case of totally geodesic foliations. Let (M,H,h) be a sub-Riemannian manifold with an integrable, metric-preserving complement V . Let v

be a chosen metric on V and assume that ∇v∗ = 0. First of all, if vol is the volumeform of the Riemannian metric g corresponding to v, then ∆′

H coincides with thesub-Laplacian ∆H defined relative to vol by Section 2.4. By Theorem 3.5 we alsoobtain a somewhat simpler curvature-dimension inequality

(CD) Γh∗+ℓv∗

2 (f) ≥ 1

n(Lf)2 +

(ρ1 −

1

ℓ

)Γh

∗

(f) + ρ2Γv∗

(f),

ρ1 = ρRicH − c−1, ρ2 = 12m

2R − cM 2

RicHV.(3.4)

where c > 0 is arbitrary. The inequality (CD) with the additional assumption ρ2 > 0was originally suggested as a generalization of the curvature-dimension inequalityby Baudoin and Garofalo [6].

We will also need the following relation, which is closely related to the inequality(CD). The proof is left to Section 3.6. This result is essential for proving the resultof Theorem 5.1 (b).

Proposition 3.7. For any f ∈ C∞(M), and any c > 0 and ℓ > 0, we have

1

4Γh

∗

(Γh∗

(f)) ≤ Γh∗

(f)(Γh

∗+ℓv∗

2 (f)− (1 − ℓ−1)Γh∗

(f)− 2Γv∗

(f)).

1

4Γh

∗

(Γv∗

(f)) ≤ Γv∗

(f)Γv∗

2 (f),

where 1 = ρRicH − c−1 and 2 = −cM 2RicHV

.


Remark 3.8. To give some context for Proposition 3.7, consider the followingspecial case. Let h = g be a complete Riemannian metric on M with lowerRicci bound ρ an choose v∗ = 0. Inserting this in (CD) with ℓ = ∞ gives us

Γg∗

2 (f) ≥ 1n (∆f) + ρΓg

∗

(f) where ∆ is the Laplacian of g. If we let Pt = et∆/2 be

the heat semi-group of 12∆, then the previously mentioned inequality implies the

inequality Γg∗

(Ptf) ≤ e−ρtPtΓg∗

(f) for any smooth, compactly supported function

f . However, Proposition 3.7 gives us Γg∗

(Γg∗

(f)) ≤ 4Γg∗

(f)(Γg

∗

2 (f)− ρΓg∗

(f))

which imply the stronger result Γg∗

(Ptf)1/2 ≤ e−ρ/2tPt(Γg

∗

(f)1/2) for any smooth,compactly supported function f , see e.g. [4, Section 2].

Remark 3.9. If a metric v on V exist with ∇v = 0, then it is uniquely determinedby its value at one point. To see this, let v′ be an arbitrary metric on V and let γbe a horizontal curve in M . Define ∇′ with respect to v′. Then by Lemma 3.2 (a),

we still have ∇′γv = 0. Finally, since H is bracket generating, the value of v at any

point can be determined by parallel transport along a horizontal curve from onegiven point.

3.4. A convenient choice of bases for H and V. Let (M,H,h) be a sub-Riemannian manifold with a metric-preserving complement of V . Let v be a metrictensor on V . To simplify the proof of Theorem 3.5, we first want to introduce aconvenient choice of bases for H and V that will simplify our calculations, similar tochoosing the coordinate vector fields of a normal coordinate system in Riemanniangeometry. Let ˚∇ be defined as in (2.6).

Lemma 3.10. Given an arbitrary point m0 of M , there are local orthonormal bases

X1, . . . , Xn and V1, . . . , Vν of respectively H and V defined in a neighborhood around

m0 such that for any vector field Y ,

(3.5) ˚∇YXi|m0=˚∇Y Vs|m0

= 0.

In particular, these bases have the properties

prH[Xi1 , Xi2 ]|m0= 0, prV [Vs1 , Vs2 ]|m0

= 0.(3.6)

Proof. Define a Riemannian metric g by g = pr∗H h+ pr∗V v. Let (x1, . . . , xn+ν) bea normal coordinate system relative to g centered at m0 such that

Hm0= span

∂

∂x1

∣∣∣∣m0

, . . . ,∂

∂xn

∣∣∣∣m0

, Vm0

= span

∂

∂xn+1

∣∣∣∣m0

, . . . ,∂

∂xn+ν

∣∣∣∣m0

.

Define Yj = prH∂∂xj

and Wj = prV∂

∂xn+j. These vector fields are linearly indepen-

dent close to m0. Write

Yj =

n+ν∑

i=1

aij∂

∂xi, Ws =

n+ν∑

i=1

bis∂

∂xi,

where

aij(m0) =

1 if i = j0 if i 6= j

, bis(m0) =

1 if i = s+ n0 if i 6= s+ n

.

Consider the matrix-valued functions

a = (aij)ni,j=1, b = (bn+r,n+s)

νr,s=1.


These matrices remain invertible in a neighborhood of m0. On this mentioned

neighborhood, let A = (Aij) = a−1 = and B = (Brs) = b−1. Define Yj =∑ni=1 AijYi and Ws =

∑νr=1BrsWr. These bases can then we written on the

form

Yj =∂

∂xj+

n+ν∑

i=n+1

aij∂

∂xi, Wj =

∂

∂xn+j+

n∑

i=1

bij∂

∂xi,

for some functions aij and bij which vanish at m0. These bases clearly satisfy (3.5)and (3.6).

Since ˚∇ preserves the metric, we can use the Gram-Schmidt process to obtain

X1, . . . , Xn and V1, . . . Vν from respectively Y1, . . . , Yν and W1, . . . , Wν .

By computing ∇ −˚∇, we obtain the following corollary.

Corollary 3.11. Given an arbitrary point m0 of M , then around m0 there are local

orthonormal bases X1, . . . , Xn and V1, . . . , Vν of respectively H and V such that for

any vector field Y ,

∇YXi|m0=

1

2♯g(Y,R(Xi, ·))|m0

,

∇Y Vs|m0= −1

2♯(∇Y g)(Vs, ·)|m0

.

where ♯ : T ∗M → TM is the identification defined relative to g. In particular, these

bases have the properties

prH[Xi1 , Xi2 ]|m0= 0, prV [Vs1 , Vs2 ]|m0

= 0.

3.5. Proof of Theorem 3.5. Let h and v be the respective metric on H and V ,which gives us a Riemannian metric g = pr∗H h+pr∗V v. Let : TM → T ∗M be the

map v 7→ g(v, ·) with inverse ♯. Let ♯v∗

be defined similar to the definition of ♯h∗

in Section 2.1. Note that ♯h∗

= prH ♯ and ♯v∗

= prV ♯.Let X1, . . . , Xn be as in Corollary 3.11 relative to some point m0. Clearly, for

any f ∈ C∞(M), we have Lf(m0) =∑n

i=1X2i f(m0). Note also that

∇Xdf(Y ) = ∇Y df(X)− df(T ∇(X,Y )) = ∇Y df(X) + df(R(X,Y )).

In the following calculations, keep in mind that ∇X♯h∗

df = ♯h∗∇Xdf , while ∇X♯

v∗

df =♯v

∗∇Xdf + (∇Xv∗)(df, ·), since V i metric-preserving.Below, all terms are evaluated at m0. We first note that for any ℓ > 0,

Γh∗+ℓv∗

2 (f) =1

2

n∑

i=1

X2i

(‖df‖2h∗ + ℓ‖df‖2v∗

)− h∗(df, dLf)− ℓv∗(df, dLf)

=

n∑

i=1

Xi∇Xidf(♯h

∗

df) + ℓ

n∑

i=1

Xi∇Xidf(♯v

∗

df)

+1

2ℓ

n∑

i=1

Xi(∇Xiv∗)(df, df) − (♯h

∗

df + ℓ♯v∗

df)

(n∑

i=1

∇Xidf(Xi)

)

=

n∑

i=1

∇Xi∇♯h∗dfdf(Xi) +

n∑

i=1

Xidf(R(Xi, ♯h∗

df))


+ ℓ

n∑

i=1

∇Xi∇♯v∗dfdf(Xi)−

n∑

i=1

∇♯h∗df ∇Xidf(Xi)

− ℓ

n∑

i=1

∇♯v∗df ∇Xidf(Xi)−

1

2ℓ

n∑

i=1

df(R(Xi, ♯h∗∇Xi

df))

+1

2ℓ(∆′

Hv∗)(df, df) + ℓn∑

i=1

(∇Xiv∗)(∇Xi

df, df).

Observe that

∇Xi∇♯h∗dfdf(Xi)− ∇♯h∗df∇Xi

df(Xi)

=

n∑

i=1

g(R∇(Xi, ♯h∗

df)♯h∗

df,Xi) +

n∑

i=1

∇[Xi,♯h∗df ]df(Xi)

= RicH(♯h∗

df, ♯h∗

df) +

n∑

i=1

∇∇Xi♯h∗dfdf(Xi) +

n∑

i=1

∇R(Xi,♯h∗df)df(Xi)

= RicH(♯h∗

df, ♯h∗

df) +

n∑

i=1

‖∇Xidf‖2h∗

−n∑

i=1


df)) +

n∑

i=1

∇Xidf(R(Xi, ♯

h∗

df)),

while

∇Xi∇♯v∗dfdf(Xi)− ∇♯v∗df ∇Xi

df(Xi)(3.7)

=n∑

i=1

g(R∇(Xi, ♯v∗

df)♯h∗

df,Xi)

+

n∑

i=1

∇∇Xi♯v∗dfdf(Xi)−

n∑

i=1

∇∇♯v

∗dfXidf(Xi)

=

n∑

i=1

∇♯v∗∇Xidf+(∇Xi

v∗)(df, ·)df(Xi)−1

2

n∑

i=1

∇♯h∗df(R(Xi, ·))df(Xi)

=

n∑

i=1

‖∇Xidf‖2v∗ +

n∑

i=1

(∇Xiv∗)(df, ∇Xi

df)

− 1

2

n∑

i=1


df)) + ‖df(R(· , ·))‖2∧2 g∗ .

Hence

Γh∗+ℓv∗

2 (f) =n∑

i=1

‖∇Xidf‖2h∗ +RicH(♯h

∗

df, ♯h∗

df)−n∑

i=1


df))

+

n∑

i=1

∇Xidf(R(Xi, ♯

h∗

df)) +

n∑

i=1

Xidf(R(Xi, ♯h∗

df))

+ ℓ

n∑

i=1

‖∇Xidf‖2v∗ + ℓ

n∑

i=1


df)


− ℓ

n∑

i=1


df)) + ℓ‖df(R(·, ·))‖2∧2 g∗

+1

2ℓ(∆′

Hv∗)(df, df) + ℓ

n∑

i=1

(∇Xiv∗)(∇Xi

df, df).

By realizing thatn∑

i=1

Xidf(R(Xi, ♯h∗

df)) =RicHV(♯df, ♯df) +n∑

i=1

∇Xidf(R(Xi, ♯

h∗

df))

+

n∑

i=1


df)),

and that


df)) = ‖df(R(· , ·))‖2∧2 g∗ ,

we obtain

Γh∗+ℓv∗

2 (f) =

n∑

i=1

‖∇Xidf‖2h∗ +RicH(♯h

∗

df, ♯h∗

df)(3.8)

+ RicHV(♯df, ♯df) + 2

n∑

i=1

∇Xidf(R(Xi, ♯

h∗

df))

+ ℓ

n∑

i=1

‖∇Xidf‖2v∗ + 2ℓ

n∑

i=1


df)

+1

2ℓ(∆′

Hv∗)(df, df).

Clearly

ℓ

n∑

i=1

‖∇Xidf‖2v∗ + 2∇Xi

df(R(Xi, ♯h∗

df)) + 2ℓ

n∑

i=1


df)

≥ −1

ℓ

n∑

i=1

‖ℓ(∇Xiv∗)(df, ·) +R(Xi, ♯

h∗

df)‖2v

≥ −M 2Rℓ

Γh∗

(f)− 2MRM∇v∗

√Γh∗(f)Γv∗(f)− ℓM 2

∇v∗Γv

∗

(f),

and alson∑

i=1

‖∇Xidf‖2h∗ =

n∑

i,j=1

(1

2(∇2

Xi,Xjf + ∇2

Xj ,Xif) +

1

2(∇Xi,Xj

f − ∇Xi,Xjf)

)2

=

n∑

i,j=1

(1

2(∇2

Xi,Xjf + ∇2

Xj ,Xif)

)2

+

n∑

i,j=1

(1

2(∇Xi,Xj

f − ∇Xi,Xjf)

)2

≥n∑

i=1

(∇2Xi,Xi

f)2 +n∑

i,j=1

(1

2df(R(Xi, Xj)))

2 ≥ 1

n(Lf)2 +

1

2m2RΓv

∗

(f).

In conclusion

Γh∗+ℓv∗

2 (f) ≥ 1

n(Lf)2 +

(ρRicH − M 2

Rℓ

)Γh

∗

(f)(3.9)


− (2MRicHV+ 2MRM∇v∗)

√Γh∗(f)Γv∗(f)

+1

2(m 2R + ℓ(ρ∆′

Hv∗ − 2M 2

∇v∗))Γv

∗

(f)

≥ 1

n(Lf)2 +

(ρRicH − 1

c− M 2

Rℓ

)Γh

∗

(f)

+1

2

(m2R − 2c(MRicHV

+ MRM∇v∗)2)Γv

∗

(f)

+ ℓ(ρ∆′

Hv∗ − 2M 2

∇v∗)Γv

∗

(f).

3.6. Proof of Proposition 3.7. LetX1, . . . , Xn be a local orthonormal basis ofH.From the assumption ∇v∗ = 0 and (3.8), we obtain Γv

∗

2 (f) =∑n

i=1 ‖∇Xidf‖2v∗ and

from this, we know

Γh∗

(Γv∗

(f)) = 4

n∑

i=1

v∗(∇Xidf, df)2

≤ 4n∑

i=1

‖∇Xidf‖2v∗‖df‖v∗ = 4Γv

∗

2 (f)Γv∗

(f).

Similarly, from (3.8),

n∑

i=1

‖∇Xidf‖2h∗ = Γh

∗+ℓv∗

2 (f)− RicH(♯h∗

df, ♯h∗

df)− 2RicHV(♯v∗

df, ♯h∗

df)

− 2

n∑

i=1

∇Xidf(R(Xi, ♯

h∗

df)) + ℓ

n∑

i=1

‖∇Xidf‖2v∗

≤ Γh+ℓv∗

2 (f)− (ρRicH − c−1 − ℓ−1)Γh∗

(f) + cM 2RicHV

Γv∗

(f),

and the result then follows.

3.7. For a general choice of L. Let (M,H,h) and V be as in Section 3.2 andlet ∆′

H be the sub-Laplacian of V . For a general choice of L with qL = h∗, writeL = ∆′

H + Z for some vector field Z. We want to use our generalized curvature-dimension inequality for ∆′

H to extend it to a more general class of operators.Unfortunately, our possibilities are somewhat limited.

Proposition 3.12. Let L = ∆′H +Z where Z ∈ Γ(V) is a non-zero vertical vector

field. Then L satisfy (CD*) with rankH < n ≤ ∞ and ρ1, ρ2,1, ρ2,1 given by

ρ1 = ρRicH − c−1,

ρ2,0 =1

2m2R − c(MZ

RicHV+ M∇v∗)

2 − 1

n− rankH‖Z‖2v∗,

ρ2,1 =1

2ρ∆′

Hv∗ − M

2∇v∗

− N2,

for any positive c > 0. Here, −MZRicHV

is a lower bound of

RicZHV(X,Y ) := RicHV(X,Y ) +1

2h(prHX, ∇pr

HY Z) +

1

2h(prH Y, ∇pr

HXZ)

+1

2v(prV X,R(Z, Y )) +

1

2v(prV Y,R(Z,X)).


and N is a lower bound of v(prV · , ∇prV ·Z). The other constants are as in Section

3.2.

Proof. Let ♯h∗

be defined as in Section 2.1 and let ♯v∗

be define analogously. Thenthe result follows from the identities,

1

2ZΓh

∗

(f)− Γh∗

(Zf, f) = df(∇♯h∗dfZ) + df(R(Z, ♯h∗

df)),

1

2ZΓv

∗

(f)− Γv∗

(Zf, f) = df(∇♯v∗dfZ) +1

2(∇Zv

∗)(df, df),

1

rankH (∆′Hf)

2 ≥ 1

n(∆′

Hf + Zf)2 − 1

n− rankH (Zf)2.

which hold for any vector field Z (not necessarily vertical).

The proof of Proposition 3.12 also shows why it is complicated to extend thisformalism to the more general case. If prH Z 6= 0, then the term ℓdf(∇♯h∗df prH Z)

requires a lower bound on the form ℓbΓh∗

(f) + ℓbΓv∗

(f) or bΓh∗

(f) + ℓ2bΓv∗

(f),both of which would be outside of our formalism.

3.8. Generalization to the case when V is not integrable. Not every vectorbundle has an integrable complement [10], let alone a metric-preserving one. Wegive a brief comment on how our results can be generalized to the case when V isnot integrable.

Let R be defined as in (3.2) and let R be defined as

R(X,Y ) := prH[prV X, prV Y ], X, Y ∈ Γ(TM).

We will adopt the terminology of [18, Ch II.8] and call R and R respectively thecurvature and the co-curvature of H. Then our theory can still be applied with thefollowing modifications.

(a) We consider a complement V as metric-preserving if

(3.10) pr∗H LV (pr∗H h) = 0, V ∈ Γ(V).Notice the difference between above formula and (3.1). In fact, (3.1) holds ifand only if (3.10) holds and V is integrable. Note that (3.10) is equivalent to

stating that ∇h∗ = 0 with respect to any connection ∇ defined as in (2.3) usingsome metric g which tames h and makes V the orthogonal complement of H.

(b) In Section 3, we now have that T ∇ = −R − R in Lemma 3.2 (c). As aconsequence, in Lemma 3.3 (b), we obtain

g(R∇(X,Y )Z −R∇(X,Z)Y,X) = g(R(Y,R(Z,X))−R(Z,R(Y,X)), X).

Hence, g(R∇(X, prH Y ) prH Z,X) = g(R∇(X, prH Z) prH Y,X), however, we

now have g(R∇(X, prV Y )Z = g(R(Y,R(Z,X)), X).

(c) In Section 3.4, Lemma 3.10 still holds, but since ˚∇ − ∇ is different, in Corol-lary 3.10 we have

∇Y Vs|m0= −1

2♯(∇Y g)(Vs, ·)|m0

+1

2♯g(Y,R(Vs, ·)).

(d) In the proof of Theorem 3.5 in Section 3.5, the only difference is that in

(3.7), we cannot be sure that the term∑n

i=1 g(R∇(Xi, ♯

v∗

df)♯h∗

df,Xi) =


trR(♯v∗

df,R(♯h∗

df, ·)) vanishes. Hence, we require the separate assumptionthat for any vector v ∈ TM on M , we have

(3.11) trR(v,R(v, ·)) = 0.

The same assumption also guarantees that

trR∇(prH · , Y )Z = trR∇(prH · , prH Y ) prH Z,

so the definition of RicH in Proposition 3.4 is still valid. If this does indeed hold,then Theorem 3.5 remains true even if V is not integrable. As a consequence,all further results in this paper and in Part II also hold in this case. Since wedo not have any geometric interpretation for the requirement (3.11), we preferto mainly consider the case when V is integrable.

The only exception are the results in Section 3.7, which do not hold when Vis not integrable even when (3.11) is satisfied.

Example 3.13. Consider the Lie algebra su(2) with basis A,B,C satisfying com-mutation relations

[A,B] = C, [A,C] = −B, [B,C] = C.

Consider its complexification, which is isomorphic to sl(2,C). Define a sub-Riemannianmanifold (SL(2,C),H,h) by considering iA, iB, iC and C as an orthonormal ba-sis for H. Here, we have used the same symbol for an element of the Lie algebraand its corresponding left invariant vector field. Then V spanned by A and B is ametric-preserving complement that is not integrable, but satisfy (3.11). Hence, allresults in this paper, with the exception of Section 3.7, hold. All results in Part IIholds as well.

4. Spectral gap and examples

4.1. The curvature-dimension inequality and a spectral gap. Let (M,H,h)be a sub-Riemannian manifold where H is bracket-generating. Let L be a smoothsecond order operator without constant term satisfying qL = h∗. Assume also thatL is symmetric with respect to some volume form vol on M . Since the metric dccinduced by h∗ is obviously complete onM , we have that L is essentially self-adjointon C∞(M) by [25, Sec 12]. Denote its (unique) self-adjoint extension to an operatoron L2(M) also by L and let Dom(L) be its domain.

Proposition 4.1. Assume that L satisfy (CD*) with ρ2,0 > 0. Let λ be any nonzero

eigenvalue of L. Then

nρ2,0n+ ρ2,0(n− 1)

(ρ1 −

k2ρ2,0

)≤ −λ, k2 = max0,−ρ2,1.

Proof. Since we know that L − λ is hypoelliptic for any λ by [16], we know thatall eigenfunctions of L are smooth. If we write 〈f, g〉 =

∫Mfg dvol, note that∫

MLf dvol = 〈Lf, 1〉 = 0 and

∫M

Γh∗

(f, g) = −〈f, Lg〉 for f, g ∈ C∞(M). Since Lis a nonpositive operator, any nonzero eigenvalue is negative. Note first that from(CD*), we get

∫

M

(Γh∗

2 (f) + ℓΓv∗

2 (f)) dvol = 〈Lf, Lf〉 − 〈Γv∗

(f, Lf), ℓ〉

≥ 1

n〈Lf, Lf〉 −

(ρ1 −

1

ℓ

)〈f, Lf〉+ 〈Γv∗

(f), ρ2,0 + ℓρ2,1〉.


Hence, if f satisfy Lf = λf , then

n− 1

nλ2〈f, f〉 ≥ −λ

(ρ1 −

1

ℓ

)〈f, f〉+ 〈Γv∗

(f), ρ2,0 + ℓ(ρ2,1 + λ)〉.

We choose ℓ =ρ2,0

−λ+k2 and obtain

n− 1

nλ2 ≥ −λ

(ρ1 −

−λ+ k2ρ2,0

),

from which the result follows.

We consider the special case of a compact sub-Riemannian manifold (M,H,h).Let g be a Riemannian metric taming h such that the orthogonal complement Vof H is integrable. Assume that ∇g = 0 where ∇ is defined as in (2.3). Then

V is a metric-preserving complement and ∇v∗ = 0 where g |V =: v. Let vol bethe volume form of g and let ∆H the sub-Laplacian of vol, which will also be thesub-Laplacian of V .

Corollary 4.2. Assume that the assumptions of Theorem 3.5 hold with k =12ρRicHm

2R − M 2

RicHV> 0. Then, for any rankH ≤ n ≤ ∞,

2k

2MRicHV+ mR

√2ρRicH + 2n−1

n k

2

≤ −λ.

Proof. This follows from the formulas (3.4) and by choosing the optimal valueof c.

4.2. Privileged metrics. Let (M,H,h) be a sub-Riemannian manifold with Hbracket-generating and regular of step r as in Remark 2.2. Let dimM = n+ν withn being the rank of H. Any Riemannian metric g on M such that g|H = h, givesus automatically a splitting TM = H⊕V1⊕ · · · ⊕Vr−1 where Vk is the orthogonalcomplement of Hk in Hk+1. Conversely, each such splitting has a canonical way ofconstructing a Riemannian metric g, which we will call privileged. Define a vectorbundle morphism

Φ: H⊕H⊗2 ⊕ · · · ⊕ H⊗r → H⊕ V1 ⊕ · · · ⊕ Vr−1,

such that component wise, Φ is the identity on the first component, while elementsin H⊗j , j ≥ 2 are sent to Vj−1 by

X1 ⊗X2 ⊗ · · · ⊗Xj 7→ prVj−1[X1, [X2[· · · [Xj−1, Xj]] · · · ]].

Give H⊗j the metric h⊗j . Then Φ induces a metric g on TM by requiring Φ|(kerΦ)⊥

is an fiberwise isometry.Assume that V = ⊕r−1

k=1Vk is an integrable metric-preserving complement. Wenormalize the vertical part of the metric by defining

v =1

2 rankV1g|V andg = pr∗H h+ pr∗V v,

Then ‖R‖∧2 g∗ ⊗g = 1 on M and hence MR = 1, while mR = 1√νif r = 2 and

0 otherwise. Furthermore, if H is of step 2 and ∇XR = 0 for any X ∈ Γ(H),


then ∇v∗ = 0 and RicHV = 0, so for the sub-Laplacian ∆H of V or equivalentlythe volume form of g, we have curvature-dimension inequality

Γh∗+ℓv∗

2 (f) ≥ 1

n(∆Hf)

2 + (ρRicH − ℓ−1)Γh∗

(f) +1

2νΓv

∗

(f),

for any ℓ > 0. As a consequence, if M is compact with ρRicH > 0 and λ is anon-zero eigenvalue of ∆H then

n

n(2ν + 1)− 1ρRicH ≤ −λ,

from Proposition 4.1.The volume form of any privileged metric taming h coincide and is called Popp’s

measure. For more details, see [1].

4.3. Sub-Riemannian manifolds with transverse symmetries. For two Rie-mann manifolds (Mj ,Hj ,hj), j = 1, 2, a sub-Riemannian isometry φ :M1 →M2 isa diffeomorphism such that φ∗h∗

2 = h∗1. The later requirement can equivalently be

written as φ∗H1 ⊆ H2 and h2(φ∗v, φ∗v) = h1(v, v) for any v ∈ H1. An infinitesimal

isometry of a sub-Riemannian manifold (M,H,h) is a vector field V such that

(4.1) LV h∗ = 0.

If V is complete with flow φt, then this flow is an isometry from M to itself forevery fixed t.

We will introduce sub-Riemannian manifolds with transverse symmetries accord-ing to the definition found in [6]. This is a special case of a sub-Riemannian manifoldwith a metric-preserving complement and consists of sub-Riemannian manifolds(M,H,h) with an integrable complement V spanned by ν linearly independent vec-tor fields V1, . . . , Vν , such that each of these vector fields are infinitesimal isometries.The subbundle V is then a metric preserving complement. If v is the metric on Vdefined such that V1, . . . , Vν forms an orthonormal basis, then ∇v∗ = 0. Hence, ancomplement spanned by transverse symmetries gives us a totally geodesic foliation.

Example 4.3. Let G be a Lie group with Lie algebra g. Define consider a principal

bundle G → Mπ→ B over a Riemannian manifold (B, qg) with G acting on the

right. An Ehresmann connection H on π is called principal if Hm · a = Hm·a. Forany such principal connection H, define h = π∗qg|H. Then G acts on (M,H,h) byisometries and hence, for each A ∈ g, the vector field σ(A) defined by

σ(A)|m =d

dt

∣∣∣∣t=0

m · expG(At),

is an infinitesimal isometry. This is then an example of a sub-Riemannian manifoldwith transverse symmetries. Let ω : TM → g be principal curvature form of H, i.e.the g-valued one-form defined by

kerω = H, ω(σ(A)) = A.

Then for any inner product 〈·, ·〉 on g, define a Riemannian metric g by g(v, v) =

qg(π∗v, π∗v) + 〈ω(v), ω(v)〉, v ∈ TM. This Riemannian metric satisfies ∇g = 0.In general, the metric g is not invariant under the group action. The latter only

hold if 〈·, ·〉 is bi-invariant inner product on g. If such an inner product exist, itinduces a metric tensor qv on the vector bundle Ad(M) → B, where Ad(M) is thequotient of M × g by the action (m,A) · a = (m · a,Ad(a−1)A). Any g-valued fromζ on M satisfying ζ(X1 · a, . . . , Xj · a) = Ad(a−1)η(X1, . . . , Xj) can be considered


as an Ad(M)-valued form on B. This includes the principal connection curvatureω and the corresponding curvature form Ω(X,Y ) = dω(X,Y ) + [ω(X), ω(Y )] =−ω(R(X,Y )).

Conversely, any section F of Ad(M) can be considered a function F : M → g

satisfying F (m ·a) = Ad(a−1)F (m). Define a connection ∇ω on Ad(M) by formula∇XF = dF (hX) and let d∇ω be the corresponding covariant exterior derivative ofAd(M)-valued forms on B. If we consider Ω as a Ad(M)-valued 2-form and δ∇ω isthe formal dual of d∇ω , then

MRicHV= sup

B‖δ∇ωΩ‖qg⊗qv.

In particular, RicHV = 0 if and only if δ∇ωΩ = 0, which is the definition of aYang-Mills connection on π.

4.4. Invariant sub-Riemannian structures on Lie groups. Let G be a Liegroup with Lie algebra g. Let G have dimension n+ ν. Choose a subspace h ⊆ g ofdimension n which generate the entire Lie algebra, and give this subspace an innerproduct. Define a vector bundle H by left translation of h. Use the inner product onh to induce a left invariant metric h on H. This gives us a sub-Riemannian manifold(G,H,h) with a left-invariant structure, i.e. G acts on the left by isometries. Thismeans that right invariant vector fields are infinitesimal isometries, however, wecannot be sure that we have a complement spanned by such vector field. This isthe case if and only if there exist a subspace k of g such that Ad(a)k is a complementof h for any a ∈ G.

Consider the special case when k is a subalgebra of g with corresponding sub-group K.

(a) Define V l by left translation of k. Then V is a complement to H, but thisis not in general spanned by infinitesimal isometries. If K is closed, H canbe considered as an Ehresmann connection on π : G → G/K, but it isnot principal in general and we cannot necessarily consider the metric h aslifted from G/K.

(b) Define Vr by right translation of k. Then Vr is spanned by infinitesimalisometries. It is a complement if and only if Ad(a)k is a complement to h

for every a ∈ G. If the later holds and K is closed, H can be considered asan Ehresmann connection on π : G→ K\G.

(c) If k is an ideal (and hence K is a normal subgroup) then V l = Vr is acomplement spanned by infinitesimal isometries.

Example 4.4 (Nilpotent Lie groups). Let g be a nilpotent Lie algebra. Define adecreasing sequence of ideals g1 ⊇ g2 ⊇ · · · ⊇ gr ⊇ gr+1 = 0 by

g1 = g, gj = [g, gj−1] for any j > 1.

Assume that g and g2 have respective dimensions n+ν and ν. Choose any subspaceh such that g = h⊕ g2. Choose an inner product on h, giving it the structure of aflat Riemannian manifold.

Let G be the simply connected Lie group of g, with closed normal subgroupG2 corresponding to the ideal g2. Define H by left translation h. Then H is anEhresmann connection on the submersion π : G → h ∼= G/G2. We lift the metricfrom h to H to obtain a left invariant metric h on H. Since h is flat, RicH = 0.Define V by right (or left) translation of g2. We will only consider the case r = 2.


We use the privileged metric g of Section 4.2, appropriately normalized so that‖R‖∧2 g∗ ⊗g = 1. Then it is clear that ∇g = 0,MRicHV

= 0 and mR take the

maximal value 1ν . Defining ∆H as the sub-Laplacian of V or equivalently the volume

form of g, we then obtain inequality

Γh∗+ℓv∗

2 (f) ≥ 1

n(∆Hf)

2 − 1

ℓΓh

∗

(f) +1

2νΓv

∗

(f),

Remark 4.5. Let g be a left invariant metric on the Lie group G, with h and v asthe respective restrictions of g to a left invariant subbundle H and its orthogonalcomplement V . Then, if V is a metric-preserving complement of (G,H,h), the

conditions of Theorem 3.5 hold, but we do not necessarily have ∇g = 0.

4.5. Sub-Riemannian manifold with several metric-preserving comple-

ments. The choice of metric preserving complement may not be unique. We givetwo examples of this.

Example 4.6. Let g be a Lie algebra such that [g, g] = g and such that there exist abi-invariant metric 〈·, ·〉 on g. Note that if g is a compact semisimple Lie algebra,these requirements are always satisfied. The term “compact Lie algebra” is hereused to mean that the Killing form is negative definite. Let ‖ad‖ be the length ofad considered as an element g∗⊗End(g) and let ‖ad‖op be the operator norm of adconsidered as a map g → End(g), both relative to the appropriate inner productsinduced by 〈· , ·〉.

Write n = dim g and let G be a Lie groups with g. Give G a Riemannian metricqg by left (or right) translation of the inner product 〈· , ·〉. Let ρG be the lowerbound of the Ricci curvature of qg. In what follows, we will always use the samesymbol for an element in a Lie algebra and the corresponding left invariant vectorfield.

(a) Define

h = (A, 12A) ∈ g⊕ g : A ∈ g.From our assumptions, we know that h + [h, h] = g ⊕ g. Define H by lefttranslation on G × G. Then H is an Ehresmann connections of the followingsubmersions

πj : G×G→ G, π1(a1, a2) = a1a−12 , π2(a1, a2) = a1.

where j = 1, 2, (a1, a2) ∈ G × G. Then the pullback of qg by either π1 or π2

gives us the same metric h when restricted to H. We can write this as

h((A, 12A), (A,

12A)

)= 〈A,A〉, A ∈ g.

The sub-Laplacian defined relative to either kerπ1∗ or kerπ∗

2 is

∆H =

n∑

i=1

(Ai,1

2Ai)

2,

where A1, . . . , An is some orthonormal basis of g.First consider, V1 = kerπ1

∗ , spanned by elements (A,A), A ∈ g, with themetric v1 given as

‖(A,A)‖v1 =2√2

‖ad‖〈A,A〉1/2.


The constants we obtain are

R((A, 12A), (B,12B) =

1

2([A,B], [A,B]), ‖R‖v1

= 1.

ρRicH = ρG, MRicHV=

1

8‖ad‖2op, M∇v∗

1= 0, mR =

‖ad‖op‖ad‖ ,

giving us the inequality

Γh∗+ℓv∗

1

2 (f) ≥ 1

n(∆Hf)

2+(ρG−c−1−ℓ−1)Γh∗

(f)+

(1

2

‖ad‖2op‖ad‖2 − c

1

64‖ad‖4op

)Γv

∗

1 (f),

for any c > 0. However, we get a better result by choosing V2 = kerπ∗, withmetric

‖(0, A)‖v2 =4√2

‖ad‖〈A,A〉1/2.

We then have

R((A, 12A), (B,12B)) = − 1

4 (0, [A,B]), ‖R‖v1= 1,

ρRicH = ρG, MRicHV= 0, M∇v∗

2= 0, mR =

‖ad‖op‖ad‖ .

so

Γh∗+ℓv∗

2

2 (f) ≥ 1

n(∆Hf)

2 + (ρG − ℓ−1)Γh∗

(f) +1

2

‖ad‖2op‖ad‖2 Γv

∗

2 (f).

From Proposition 4.1, we conclude that if λ is a nonzero eigenvalue of ∆H, then

n‖ad‖opn(‖ad‖op + 2‖ad‖)− ‖ad‖op

ρG ≤ −λ.

(b) Let I : g → Rn be a bilinear map of vector space. Define h as a subspace of

g×Rn by (A, I(A)), A ∈ g×R

n. Consider g×Rn as the Lie algebra of G×R

n

and define H by left translation of h. This is an Ehresmann connection relativeboth projections

π1 : G× Rn → G, π2 : G× R

n → Rn .

Give G the metric qg and give Rn a flat metric by the inner product 〈I(·), I(·)〉.Pulling back these metrics through respectively π1 and π2, we obtain the samesub-Riemannian metric h on H given by

‖(A, I(A))‖2h = 〈A,A〉,even though the geometry of G and R

n are very different. The sub-Laplacianwith respect both V1 = kerπ1 and V2 = kerπ2 also coincides and is given by

∆H =

n∑

i=1

(A, I(A))2, A1, A2, . . . , An is an orthonormal basis of g.

We will leave out most of the calculations, and only state that if we definevj on Vj = kerπj∗ by

‖(0, I(A)‖2v1= 〈A,A〉, ‖(A, 0)‖2v1

= 〈A,A〉,then we have inequalities

Γh∗+ℓv∗

1

2 (f) ≥ 1

n(∆Hf)

2 + (ρG − c−1 − ℓ−1)Γh∗

(f) +1

4

(2‖ad‖op‖ad‖ − ‖ad‖2op

)Γv

∗

1 (f),


Γh∗+ℓv∗

2

2 (f) ≥ 1

n(∆Hf)

2 − (c−1 + ℓ−1)Γh∗

(f) +1

4

(2‖ad‖op‖ad‖ − ‖ad‖2op

)Γv

∗

2 (f).

which hold for any c > 0 and ℓ > 0.

4.6. A non-integrable example. As usual, we use the same symbol for an ele-ment of the Lie algebra and the corresponding left invariant vector field. Considerthe complexification of su(2) spanned over C by

[A,B] = C, [A,C] = −B, [B,C] = A.

Consider su(2)C as the Lie algebra of SL(2,C) and on that Lie group, define (real)subbundles of the tangent bundle

H = spaniA, iB, iC,C, V = spanA,B.Define a Riemannian metric g such that iA, iB, iC,C,

√2A and

√2B forms an

orthonormal basis. Define g |H = h and g |V = v and let ∆′H be the sub-Laplacian

of the sub-Riemannian manifold (H,h) with respect to the complement V . It issimple to verify that

∆′H = (iA)2 + (iB)2 + (iC)2 + C2.

Let R and R be respectively the curvature and the co-curvature of H. Note that

∇h∗ = 0, ∇v∗ = 0.

trR(v,R(v, ·)) = 0, for any v ∈ TM.

RicH(Y, Y ) = −5

2g(iA, Y )2 − 5

2g(iB, Y )2 − 2 g(iC, Y )2 +

1

2g(C, Y )2.

RicHV = 0, ‖R‖g⊗∧2g∗ = 1.

It follows that ∆′H is also the sub-Laplacian of the volume form of g with curvature-

dimension inequality

Γh∗+λv∗

2 (f) ≥ 1

4(∆′

Hf)2 −

(5

2+

1

λ

)Γh

∗

(f) +1

4Γv

∗

(f).

5. Summary of Part II

We include a section here to illustrate what further results can be obtained fromour curvature-dimension inequality (CD*) of Theorem 3.5.

Let L be a second order operator. Let X(m) be an L-diffusion with X(m) =m and maximal lifetime τ (m). For any bounded f , define Ptf by Ptf(m) =E[f(Xt(m))1t≤τ ]. Assume that L satisfy an inequality (CD*) with respect to some

v∗ ∈ Γ(Sym2 TM). Let C∞b (M) denote the space of all smooth, bounded functions.

We will introduce two important conditions.

(A) Pt1 = 1 and for any f ∈ C∞b (M) with Γh

∗+v∗

(f) ∈ C∞b (M) and for every

T > 0, we have

supt∈[0,T ]

‖Γh∗+v∗

(Ptf)‖L∞ <∞.

(B) For any f ∈ C∞(M), we have Γh∗

(f, Γv∗

(f)) = Γv∗

(f, Γh∗

(f)).

We have the following concrete classes of sub-Riemannian manifolds satisfying theseconditions.

Theorem 5.1. Part II, Proposition 3.3, Theorem 3.4.


(a) Assume that π : M → B is a fiber-bundle with compact fibers over a Riemannian

manifold (B, qg). Let H be an Ehresmann connection on π and define h =π∗qg|H. Assume that the metric dcc of (M,H,h) is complete and that the sub-

Laplacian ∆′H of V = kerπ∗ satisfy (CD*). Then condition (A) holds.

(b) Let M be a sub-Riemannian manifolds with an integrable, metric preserving

complement V. Assume that there exist a choice of v on V such that g =pr∗H h+ pr∗V v is a complete Riemannian metric and such that ∇v∗ = 0. Then

the sub-Laplacian ∆H of V and the volume form of vol of g coincide. Assume

that ∆H satisfy the assumptions of Theorem 3.5. Finally, assume that mR > 0.Then conditions (A) and (B) hold.

The result of Theorem 5.1(b) is also valid in the non-integrable complements ofthe type described in Section 3.8.

For the special case when h∗ is a complete Riemannian metric, v∗ = 0 and Lis the Laplacian, (A) always hold when the curvature-dimension inequality holds.For this reason, we expect that the condition (A) will hold in more cases than theones listed here.

Combined with our generalized curvature-dimension inequality, we have the fol-lowing identities.

Theorem 5.2. (Part II, Proposition 3.8) Assume that L satisfy (CD*) with respect

to v∗ and that (A) holds. Assume also that L is symmetric with respect to the

volume form vol, i.e.∫M fLg dvol =

∫M gLf dvol for any f, g ∈ C∞

c (M). Finally,

assume that g∗ = h∗ + v∗ is the co-metric of a complete Riemannian metric.

(a) Assume that ρ1 ≥ ρ2,1 and that ρ2,0 > −1. Then for any compactly supported

f ∈ C∞c (M),

‖Γh∗

(Ptf)‖L1 ≤ e−αt‖Γh∗

(f)‖L1 , α :=ρ2,0ρ1 + ρ2,1ρ2,0 + 1

.

Furthermore, if α > 0 then vol(M) <∞.

(b) Assume that the conditions in (a) hold with α > 0. Then for any f ∈ C∞c (M),

‖f − fM‖2L2 ≤ 1

α

∫

M

Γh∗

(f) dvol .

where fM = 1vol(M)

∫M f dvol. As a consequence if λ is a non-zero eigenvalue

of L, then α ≤ −λ.

There are also other inequalities which do not require that L is symmetric withrespect to a volume form, see e.g. Part II, Proposition 3.6.

For the case when both (A) and (B) hold, and where L satisfy (CD) of Section 3.3with ρ2 > 0, we can use the results from [6, 7, 8]. In particular, with some extracomputation, we can conclude the following.

Corollary 5.3. (Part II, Proposition 5.1, Proposition 5.3) Let ∆H is as in Theo-rem 5.1 (b). Let

k = 12mRρRicH − M

2RicHV

.

(a) Assume that k > 0. Then M is compact and for any f ∈ C∞(M)

‖f − fM‖2L2 ≤ 1

α

∫

M

Γh∗

(f) dvol .


where fM = 1vol(M)

∫Mf dvol and

α :=

(2k

2MRicHV+ mR

√2ρRicH + 2k

)2

.

(b) Assume that k ≥ 0. Let pt(m0,m1) be the heat kernel of 12∆H with respect to

vol. Then

pt(m,m) ≤ 1

tN/2p1(m,m).

where m ∈M, 0 ≤ t ≤ 1 and

N =n

4

(√2ρRicH + k +

√ρRicH + k

)2

k.

Furthermore, for any 0 < t0 < t1, any non-negative smooth bounded function

f ∈ C∞b (M) and points m0,m1 ∈M

Pt0f(m0) ≤ (Pt1f)(m1)

(t1t0

)N/2exp

(√(k + ρRicH)(k + 2ρRicH)

2k (t1 − t0)d(m0,m1)

2

).

If k = 0, we interpret the quotient k

ρRicH

as 12m

2R.

References

[1] A. Agrachev, U. Boscain, J. P. Gauthier and F. Rossi, The intrinsic hypoelliptic Laplacian

and its heat kernel on unimodular Lie groups. Journal of Functional Analysis, vol. 256 no. 8,(2009) 2621–2655.

[2] A. Agrachev, D. Barilari and L. Rizzi, The curvature: a variational approach, arXiv:1306.5318v3

[3] D. Bakry and M. Emery, Diffusions hypercontractives. Seminaire de probabilites, XIX, 177–206, Lecture Notes in Math., 1123, Springer, Berlin, 1985.

[4] D. Bakry and M. Ledoux, Levy-Gromov’s isoperimetric inequality for an infinite-dimensional

diffusion generator. Invent. Math. vol. 123 , no. 2 (1996), 259–281.[5] D. Barilari and L. Rizzi Comparison theorems for conjugate points in sub-Riemannian ge-

ometry, arXiv:1401.3193.[6] F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for

sub-Riemannian manifolds with transverse symmetries arXiv:1101.3590.[7] F. Baudoin and M. Bonnefont, Log-Sobolev inequalities for subelliptic operators satisfying a

generalized curvature dimension inequality. J. Funct. Anal. vol. 262 (2012), 2646–2676.[8] F. Baudoin, M. Bonnefont and N. Garofalo, A sub-Riemannian curvature-dimension in-

equality, volume doubling property and the Poincar inequality. Math. Ann. vol. 358, no. 3-4(2014), 833–860.

[9] F. Baudoin and J. Wang, Curvature Dimension Inequalities and Subelliptic Heat Kernel

Gradient Bounds on Contact Manifolds. Potential Anal. vol. 40, no. 2 (2014), 163–193[10] R. Bott, On a topological obstruction to integrability. 1970 Global Analysis (Proc. Sympos.

Pure Math., Vol. XVI, Berkeley, Calif., 1968), 127–131 Amer. Math. Soc., Providence, R.I.

[11] W. L. Chow, Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung.

Math. Ann. 117 (1939), 98–105.[12] B.K. Driver and T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group.

Journal of Functional Analysis, vol. 221, no. 2 (2005), 340–365.[13] M. Emery, Stochastic calculus in manifolds, Springer-Verlag, Berlin, 1989[14] M. Gromov, Carnot-Carathodory spaces seen from within, Sub-Riemannian geometry. Progr.

Math., 144, Birkhuser, Basel, 1996, 79–323.

[15] R.K. Hladky, Bounds for the first eigenvalue of the horizontal Laplacian in positively curved

sub-Riemannian manifolds. Geom. Dedicata vol. 164 (2013), 155–177.[16] L. Hormander, Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147–

171.

http://de.arxiv.org/abs/1401.3193

http://de.arxiv.org/abs/1101.3590


[17] E.P. Hsu, Stochastic analysis on manifolds. Graduate Studies in Mathematics, 38. AmericanMathematical Society, Providence, RI, 2002

[18] I. Kolar, P. W. Michor and J. Slovk,, Natural operations in differential geometry. Springer-Verlag, Berlin, 1993.

[19] C. Li and I. Zelenko, Jacobi equations and comparison theorems for corank 1 sub-Riemannian

structures with symmetries. Journal of Geometry and Physics, vol. 61, no. 4 (2011), 781–807.[20] P.A. Meyer, Geometrie stochastique sans larmes, I. Seminare de probabilites de Strasbourg,

15 (1981)[21] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math-

ematical Surveys and Monographs, 91. American Mathematical Society, Providence, RI, 2002.[22] P.K. Rashevskiı, About connecting two points of a completely nonholonomic space by admis-

sible curve, Uch. Zapiski Ped. Inst. K. Liebknecht vol. 2 (1938), 83–94.[23] B. L. Reinhart, Foliated manifolds with bundle-like metrics, The Annals of Mathematics, vol.

69 (1959), 119–132.[24] I. Shigekawa, On stochastic horizontal lifts, Z. Wahrsch. Verw. Gebiete, vol 59 (1982), no. 2,

211–221.[25] R. S. Strichartz, Sub-Riemannian geometry. Journal of Differential Geometry, vol. 24, no. 2,

(1986) 221–263.[26] K.-T. Sturm, Analysis on local Dirichlet spaces I. Recurrence, conservativeness and Lp-

Liouville properties. J. Reine Angew. Math. vol 456 (1994), 173–196[27] F.-Y. Wang, Equivalence of dimension-free Harnack inequality and curvature condition. In-

tegral Equations Operator Theory vol. 48, no. 4 (2004), 547–552.[28] I. Zelenko and C. Li, Parametrized curves in Lagrange Grassmannians. Comptes Rendus

Mathematique, vol. 345, no. 11 (2007), 647–652.

Mathematics Research Unit, University of Luxembourg, 6 rue Richard Coudenhove-

Kalergi, L-1359 Luxembourg

E-mail address: [email protected]

Mathematics Research Unit, University of Luxembourg, 6 rue Richard Coudenhove-

Kalergi, L-1359 Luxembourg

E-mail address: [email protected]

curvature-dimension inequalities on sub-riemannian manifolds obtained from riemannian foliations:...

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