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Home , Ricci curvature

arXiv:1711.08382v2 [math.DG] 14 Feb 2018 ihpstv ic uvtr a rtd hmchmlg g Rham de first a has Ricci positive with nlsso aiod lk og n ne hoy.Moreov that implies theory). technique developmen index exciting Bochner and most the Hodge the positive, (like of some of of origin analysis the at compute been be can has that quantity curvature a connecting result, fBcnrstcnqei h otx fattlygoei R totally a of conn context t simply the In any in sphere. that technique a that Bochner’s to [10] diffeomorphic of in is operator shown curvature was positive it flow, Ricci using eertdBcnrstcnqe[,3]yed htacompac a that yields 31] [9, technique Bochner’s Celebrated Introduction 1 oiotldistribution horizontal hc a eudrto stechmlg ftela pc.I space. leaf interes the of are cohomology authors the the as works [2, understood those instance be of can for which all (see extensive in However, is literature [30]). the and new not ∗ † uhrspotdi atb h eerhCuclo owy(p Norway of Council Research 1660031 the DMS by Grant part NSF in the supported by Author part in supported Author h td fchmlgclpoete fReana foliat Riemannian of properties cohomological of study The rnvreWizn¨c omlsadd hmchmlg of cohomology Rham de Weitzenb¨ock and Transverse formulas peri h daai ii ftecnnclvraino h metric. the of variation canonical the leave of the limit Rh to adiabatic de transverse the the quantities in for curvature appear theorems of nullity positivity obtain the we only consequence, a As foliation. epoetases etebokiette o h oiotlL horizontal the Weitzenb¨ock for transverse identities prove We eateto ahmtc,Uiest fCnetct USA Connecticut, of University , of Department ,reJlo-ui,912GfsrYet,Fac,and France, Gif-sur-Yvette, 91192 Joliot-Curie, rue 3, nvriyo egn eateto Mathematics, of Department Bergen, of University F ⊥ sbaktgenerating. bracket is nvri´ ai-u,LSS-SUP Universit´e Paris-Sud, ..Bx70,52 egn Norway Bergen, 5020 7803, Box P.O. oal edscfoliations geodesic totally [email protected] [email protected] arc Baudoin Fabrice redGrong Erlend Abstract H dR k 1 ( M for 0 = ) † ∗ oetnme 249980/F20). number roject oal otetplg ftemanifold, the of topology the to locally d retdReana aiod( Riemannian oriented t 9 2 4 5 n h ilorpyin bibliography the and 25] 24, 22, 19, e ntes-aldbsccohomology, basic so-called the in ted epeetppr esuyanalogues study we paper, present he r ftecraueoeao tefis itself operator curvature the if er, eana oito ( foliation iemannian ELEC, u ok eaerte interested rather are we work, our n ´ k so h 0hcnuyi h global the in century 20th the of ts osb h ohe’ ehiu is technique Bochner’s the by ions roup 0 6= ce imninmnfl with manifold Riemannian ected .Toecrauequantities curvature Those s. pain fattlygeodesic totally a of aplacians , mchmlg assuming cohomology am H dim dR 1 ( M M e 2,1] nfact, In 16]. [23, see , .Ti beautiful This 0. = ) M g, , F whose ) M g , ) in the full de Rham cohomology of the manifold. It should be noted that obtaining informations about the de Rham cohomology from horizontal curvature quantities only, requires the bracket- generating condition for the horizontal distribution. Unlike [24, 25] the Laplacians we consider will therefore be hypoelliptic but not elliptic. We work in a general framework of totally geodesic Riemannian foliations, but we emphasize that our results are essentially new even in the context of contact manifolds. In fact, to provide the reader with an initial motivation for our more general results, we begin with stating our main objective in such setting. Theorem 1.1. Let (M, g) be a compact, connected, oriented K-contact manifold with positive 1 M M horizontal Tanno Ricci curvature, then HdR( ) = 0. If moreover has a positive horizontal k M M Tanno curvature operator then HdR( ) = 0 for 0

Our goal in this section is to prove the following generalization of Hodge theorem: Theorem 2.1. Let M be a smooth, connected, oriented and compact manifold . We assume that M is equipped with a Riemannian foliation with bundle-like metric g and totally geodesic leaves. F 1 M We also assume that the tensor Q defined in (4) is positive. Then, HdR( ) = 0. The various assumptions of the theorem are explained in this section. We mention that in the contact case, the theorem yields:

2 Corollary 2.2. Let M be a smooth, connected, oriented and compact K-contact manifold with 1 M positive horizontal Tanno Ricci curvature, then HdR( ) = 0.

2.1 Transverse Weitzenb¨ock formulas on one-forms We first recall the framework and notations of [3, 6], to which we refer for further details. Let M be a smooth, oriented, connected, compact manifold with n + m. We assume that M is equipped with a Riemannian foliation with a bundle-like complete metric g (see [28]) and F totally geodesic m-dimensional leaves. The distribution formed by vectors tangent to the leaves V is referred to as the set of vertical directions. The distribution which is normal to is referred H V to as the set of horizontal directions. We will always assume in this paper that the horizontal distribution is everywhere step two bracket-generating. H Example 2.3. Let (M,θ) be a 2n + 1-dimensional smooth contact manifold. There is a unique smooth vector field Z, the so-called Reeb vector field, that satisfies

θ(Z) = 1, Z θ = 0, L where Z denotes the with respect to Z. On M there is a foliation, the Reeb foliation, L whose leaves are the orbits of the vector field Z. As it is well-known (see [29]), it is always possible to find a Riemannian metric g and a (1, 1)-tensor field J on M so that for every vector fields X,Y

g(X,Z)= θ(X), J 2(X)= X + θ(X)Z, g(JX,Y )= dθ(X,Y ). − The triple (M,θ,g) is called a contact . It is well-known ([29]) that the Reeb foliation is totally geodesic with bundle like metric if and only if the Reeb vector field Z is a Killing field, that is, Zg = 0. L In that case, (M,θ,g) is called a K-contact Riemannian manifold. Observe that the horizontal distribution is then the kernel of θ and that is bracket generating because θ is a contact form. H H Sasakian manifolds are examples of K-contact manifolds. We refer to [11] for further details on Sasakian foliations. The reference on M (see [2, 3, 20]), the Bott connection, is given as follows:

g π ( X Y ),X,Y Γ∞( ) H ∇ ∈ H π ([X,Y ]), X Γ∞( ),Y Γ∞( ) X Y = H ∈ V ∈ H ∇  π ([X,Y ]), X Γ∞( ),Y Γ∞( )  V g ∈ H ∈ V π ( X Y ),X,Y Γ∞( ) V ∇ ∈ V  where g is the Levi-Civita connection and π (resp. π ) the projection on (resp. ). It is easy ∇  H V H V to check that this connection satisfies g = 0. The Bott connection has a torsion which is given ∇ by: T (X,Y )= π ([X,Y ]). − V Example 2.4. Let (M,θ,g) be a K-contact manifold. The Bott connection coincides with the Tanno’s connection that was introduced in [29] and which is the unique connection that satisfies: 1. θ = 0; ∇ 2. Z = 0; ∇

3 3. g = 0; ∇ 4. T (X,Y )= dθ(X,Y )Z for any X,Y Γ ( ); ∈ ∞ H 5. T (Z, X) = 0 for any vector field X Γ ( ). ∈ ∞ H In the case where (M,θ,g) is Sasakian, the Bott connection coincides then with the Tanaka-Webster connection (see [12]).

We define the horizontal gradient f of a smooth function f as the projection of the Rieman- ∇H nian gradient of f on the horizontal bundle. Similarly, we define the vertical gradient f of a ∇V function f as the projection of the Riemannian gradient of f on the vertical bundle. The horizontal Laplacian ∆ is the generator of the symmetric pre-Dirichlet form H M (f, g)= f, g dµ, f, g C0∞( ), EH Mh∇H ∇H iH ∈ Z where µ is the Riemannian volume measure. We have therefore the following integration by parts formula M f, g dµ = f∆ gdµ = g∆ fdµ, f,g C0∞( ). Mh∇H ∇H iH − M H − M H ∈ Z Z Z From this convention ∆ is therefore non-positive. The Riemannian metric g can be split as H g = g g , H ⊕ V and for later use, we introduce its canonical variation, which is the the one-parameter family of Riemannian metrics defined by: 1 gε = g g , ε> 0. H ⊕ ε V The limit ε 0 is the sub-Riemannian limit and the limit ε + the so-called adiabatic limit. → → ∞ Remark 2.5. The canonical variation of the metric of a totally geodesic foliation has been studied for a long time (see Chapter 9 in [8] and the references therein). A prototype example is given by the canonical variation of the standard metric g on the odd-dimensional sphere S2n+1 foliated by 2n+1 1 the Hopf fibration. The Riemannian manifold (S , gε) is then called the Berger sphere. When 2n+1 2n+1 ε 0, in the Gromov-Hausdorff sense, (S , gε) converges to S endowed with the Carnot- → 2n+1 n Carath´eodory metric. When ε , (S , gε) converges to the complex projective space CP → ∞ endowed with the standard Fubini-Study metric.

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

g (JZ X,Y )= g (Z, T (X,Y )). (1) H V where T is the of . We then extend JZ to be 0 on x. Also, if Z Γ ( ), from ∇ V ∈ ∞ H (1) we set JZ = 0. m If Z1,...,Zm is a local vertical frame, the operator ℓ=1 JZℓ JZℓ does not depend on the choice of the frame and shall concisely be denoted by J2. P Example 2.6. If M is a K-contact manifold equipped with the Reeb foliation, then J is the almost complex structure on the horizontal bundle, and therefore J2 = Id . − H 1For Marcel Berger (1927-2016).

4 The horizontal divergence of the torsion T is the (1, 1) tensor which is defined in a local horizontal frame X1,...,Xn by n

δ T (X)= ( Xj T )(Xj, X). H − ∇ j X=1 The g-adjoint of δ T will be denoted δ∗ T . The foliation (M, , g) is said to be Yang-Mills if H F δ T = 0 (see [3] or [8]). H H Example 2.7. If M is a K-contact manifold, then the Reeb foliation is Yang-Mills (see [1]).

By declaring a one-form to be horizontal (resp. vertical) if it vanishes on the vertical bundle V (resp. on the horizontal bundle ), the splitting of the H

TxM = x x H ⊕V gives a splitting of the cotangent space. The metric gε induces then a metric on the cotangent bundle which we still denote gε. By using similar notations and conventions as before we define M pointwisely for every η in Tx∗ , 2 2 2 η ε = η + ε η . k k k kH k kV

By using the duality given by the metric g, (1, 1) can also be seen as linear maps on the cotangent bundle T M. More precisely, if A :Γ (T M) Γ (T M) is a (1, 1) tensor, we will often ∗ ∞ → ∞ still denote by A the fiberwise on the cotangent bundle which is defined as the g-adjoint of the dual map of A. Namely A : Γ (T M) Γ (T M) is such that for any η,ξ Γ(T M), ∞ ∗ → ∞ ∗ ∈ ∗ Aη,ξ = ξ(A♯η) where ♯ is the standard musical isomorphism. The same convention will be made h i for any (r, s) tensor. As a convention, unless explicitly mentioned otherwise in the text, the inner product duality will always be understood with respect to the reference metric g (and not gε).

We define then the horizontal Ricci curvature Ric of the Bott connection as the fiberwise H symmetric linear map on one-forms such that for every smooth functions f, g,

Ric (df), dg = Ric( f, g), h H i ∇H ∇H where Ric is the Ricci curvature of the connection . ∇ Remark 2.8. If the foliation comes from a submersion, then Ric is simply the horizontal lift of H the Ricci curvature of the leaf space.

The adjoint connection of the Bott connection is not metric. For this reason, we shall rather make use of the following family of connections first introduced in [4, 6]:

ε 1 Y = X Y T (X,Y )+ JY X, 0 < ε + ∇X ∇ − ε ≤ ∞ and we shall only keep the Bott connection as a reference connection. It is readily checked that ε ε gε = 0 when ε< + . The adjoint connection (see Appendix A) of is then given by ∇ ∞ ∇ ε 1 ˆ Y = X Y + JX Y, ∇X ∇ ε

5 thus ˆ ε is also a metric connection. It moreover preserves the horizontal and vertical bundle. If η ∇ is a one-form, we define the horizontal gradient in a local adapted frame of η as the (0, 2) tensor

n

η = Xi η θi. ∇H ∇ ⊗ i X=1 where θi, i = 1,...,n is the dual of Xi. Finally, for ε> 0, we consider the following operator which is defined on one-forms by

ε ε 1 1 2 ∆ ,ε = ( )∗ + δ T J Ric , (2) H − ∇H ∇H ε H − ε − H

2 where the adjoint is understood with respect to the (L , gε) product on sections, i.e. , εdµ. ∗ Mh· ·i It is easily seen that, in a local horizontal frame, R n ε ε ε 2 ε ε ( )∗ = ( Xi ) Xi . (3) − ∇H ∇H ∇ −∇∇Xi i X=1 2 Remark 2.9. We observe that ∆ ,ε is a symmetric operator with respect to the (L , gε) product H on sections if and only if δ T = 0, that is if and only if the foliation is of Yang-Mills type. H It is proved in [6] that for every smooth one-form α on M, the following holds

lim ∆ ,εα =∆ , α, ε H H ∞ →∞ where, in a local horizontal frame

n 2 ∞ ∆ , = ( X∞i ) ∞ Xi Ric . H ∞ ∇ −∇∇Xi − H i X=1 Our first result is the following transverse Weitzenb¨ock formula on one-forms.

Proposition 2.10. Let 0 < ε + . On one-forms, one has ≤ ∞

∆ ,ε = dδ ,ε δ ,εd, H − H − H ε where δ ,ε is the horizontal divergence for the connection defined for a p-form ω as the p 1 H ∇ − form: n ε δ ,εω(V1, ,Vp 1)= X ω(Xi,V1, ,Vp 1), H · · · − − ∇ i · · · − i X=1 in a local orthonormal horizontal frame X , , Xn. 1 · · · Proof. The proposition will be proved in greater generality in Proposition 3.3. We give here a sketch of direct proof that allows to identify explicitly the tensors on 1-forms and comparison to [6]. Let x M. From [6], around x, there exist a local orthonormal horizontal frame X , , Xn ∈ { 1 · · · } and a local orthonormal vertical frame Z , ,Zm such that the following structure relations { 1 · · · } hold n m k k [Xi, Xj ]= ωijXk + γijZk Xk=1 Xk=1

6 m j [Xi,Zk]= βikZj, j X=1 k k j where ωij, γij, βik are smooth functions such that:

βj = βk . ik − ij Moreover, at x, we have k k ωij = 0, βij = 0.

We denote by θ , ,θn the dual of X , , Xn and ν , ,νm the dual basis of Z , ,Zm. 1 · · · 1 · · · 1 · · · 1 · · · We now use Fermion calculus on one-forms and introduce the following operators acting on one- forms: a∗α = θi α, aiα = ιX α, b∗α = νl α, blα = ιZ α. i ∧ i l ∧ l We perform the following computations at the center x of this frame. It is easy to see that the d on one-forms can then be written:

1 l d = a∗ X + b∗ Z + γ a∗a∗bl i ∇ i l ∇ l 2 ij i j i X Xl Xi,j,l and that 1 l l δ ,ε = ai Xi + γjiai∗bl∗aj γjiai∗ajbl H − ∇ ε − i X Xi,j,l We can then proceed by direct computations, as in the proof of Proposition 3.3 and use Lemma 3.2 in [6] to identify the term 1 δ T 1 J2 Ric . ε H − ε − H One recovers then immediately a result first proved in [6]. Corollary 2.11. For every f C (M) and every 0 < ε + , one has the following commuta- ∈ ∞ ≤ ∞ tion: d∆ f =∆ ,εdf. H H Proof. On functions, we have for every 0 < ε + , ≤ ∞ ∆ = δ ,εd. H − H So, the result follows from Proposition 2.10 and the fact that d2 = 0.

2.2 on one-forms

2 1 2 In this section, we collect several analytic prerequisites. We will denote by L ( M, gε) the L 2 ∧ space of one-forms with respect to the (L , gε) product on sections.

Proposition 2.12. Let 0 < ε + . The operator ∆ ,ε is hypoelliptic. ≤ ∞ H Proof. We can locally write n ε 2 ∆ ,ε = ( X ) + V H ∇ i i X=1 where V is a first order real differential operator and X , , Xn a local horizontal frame that 1 · · · satisfies the two-step H¨ormander’s bracket generating condition. From H¨ormander’s theorem (see [21] for the scalar case and for instance the arguments of Proposition 3.5.1 in [27] for the form case), one deduces the hypoellipticity of ∆ ,ε. H

7 Proposition 2.13. Let 0 <ε< + . The operator ∆ ,ε is the generator of a strongly continuous ∞ 2H 1 t∆H,ε and smoothing semigroup of bounded operators on L ( T ∗M, gε). Moreover, if e ,t 0 ∧ t∆ { ≥ } denotes this semigroup, then for every smooth one-form α, αt = e H,ε α is the unique solution of the heat equation:

∂αt =∆ ,εαt, ∂t H (α0 = α.

Proof. We use arguments developed in [7, 24, 27]. First, since the manifold M is supposed to be compact, one deduces from (3) that ∆ ,ε satisfies a G˚arding type inequality: there exists a constant H Kε such that for every smooth one-form

∆ ,εα, α L2,g e Kε α L2,g . h H i v ≤ k k ε

As a consequence, ∆ ,ε is the generator of a strongly continuous semigroup of bounded operators t∆ 2 1 H e H,ε α on L ( M, gε) (see [26] pages 14 and 15 and Corollary page 81) that weakly solves the heat ∧ t∆ equation. Since ∆ ,ε is hypoelliptic, e H,ε actually admits a smooth heat kernel (as in chapter 5 H of [27]) and thus et∆H,ε also strongly solves the heat equation. It remains to prove that solutions of the heat equation are unique. So, let αt be a strong solution of

∂αt =∆ ,εαt, ∂t H (α0 = 0.

Since M is compact the functional 2 φ(t)= αt gε dµ M k k Z is well defined and one has

φ′(t) = 2 ∆ ,εαt, αt gε dµ 2Kεφ(t). Mh H i ≤ Z So from Gronwall’s lemma φ(t) = 0 and solutions of the heat equation are therefore unique.

It should be noted that this semigroup et∆H,ε α coincides with the semigroup constructed in [18] (or [4, 6] in the Yang-Mills case) and therefore admits a Feynman-Kac type representation. The next result is then an easy consequence of Corollary 2.11 and Proposition 2.13 (it is also pointed out in [4, 6, 18]).

Lemma 2.14. Let 0 <ε< + . For every t 0, and f C (M), ∞ ≥ ∈ ∞ det∆H f = et∆H,ε df

Proof. Both sides are solutions of the heat equation

∂αt =∆ ,εαt, ∂t H (α0 = df.

8 H1 M 2.3 Horizontal semigroup and dR( ) In this section, in addition of the assumptions of the previous section, we now assume a positive curvature condition. if α is a one-form, we define a tensor Q by

1 2 Q(α), α g = Ric (α), α δ T (α), α Tr (Jα) (4) h i h H iH − h H iV − 4 H and assume in this section that Q is positive in the sense that there exists a positive constant λ> 0 such that for every smooth and horizontal one-form α,

2 Q(α), α g λ α . (5) h i ≥ k kg We note that Q is only symmetric if the Yang-Mills condition δ T = 0 is satisfied. Our goal is to H 1 M prove that, if ε is large enough, harmonic forms for ∆ ,ε are zero and deduce then that HdR( ) = 0. H The main tool is the heat semigroup et∆H,ε , t 0, that was constructed in the previous subsection. ≥ Remark 2.15. The tensor Q was first introduced in [5] in the context of sub-Riemannian manifolds with transverse symmetries, and with the notations of [5], one actually has (f,f)= Q(df), df g. R h i A direct computation (as Theorem 9.70, Chapter 9 in [8]) shows that the Ricci curvature (for the Levi-Civita connection) of gε is given by,

1 Ric (v, w) 2 tr JvJw, if v, w , V − 4ε H ∈V  1 Ricgε (v, w)=  2ε δ T (v), w g if v , w ,  − h H i ∈H ∈V  1 2 Ric (v, w)+ J w, v g if v, w  H 2ε h i ∈H  with Ric being the Ricci curvature of the leafs of the foliation. In particular, we see that for every u , vV , ∈H ∈V 1 2 2 Ricgε (v + εw, v + εw)= J v, v g + Q(v + w), v + w g + ε Ric (w, w). (6) 2εh i h i V Observe that, due to the vertical curvature term Ric , our positivity assumption on Q does not V necessarily imply that there exists ε > 0 such that Ricgε > 0. Therefore Theorem 2.1, is not a consequence of the classical Hodge theorem, since we do not require any assumption on Ric . V Remark 2.16. The condition (5) can be simplified if the foliation is of Yang-Mills type, that is δ T = 0. Indeed, if Z is a vertical vector field and X1, , Xn a local horizontal frame, it is easy H · · · to compute that n n 2 2 2 Tr (JZ )= T (Xi, Xj),Z = [Xi, Xj],Z . − H h i h i i,j i,j X=1 X=1 Therefore, the step-two generating condition on and the compactness of M then implies that there H exists a positive constant a> 0 such that

1 2 2 Tr (Jα) a α . −4 H ≥ k V k Therefore, on Yang-Mills foliations (like K-contact foliations), the condition (2) is equivalent to the fact that the Ricci curvature of the Bott connection is positive on the horizontal bundle.

9 We now turn to the proof of Theorem 2.1. We have the following first lemma:

Lemma 2.17. There exist ε> 0 and a constant cε > 0 such that for every closed and smooth-one form α,

1 2 2 ∆ α ε ∆ ,εα, α ε cε α ε. (7) 2 Hk k − h H i ≥ k k Proof. As it has been shown in [6] (see also Theorem 4.7 in [3]), the formula (2) implies that for any smooth form α,

1 2 ε 2 1 2 ∆ α ε ∆ ,εα, α ε = α ε + Ric (α), α δ T (α), α + J (α), α 2 Hk k − h H i k∇H k h H iH − h H iV ε h iH A computation similar to the proof of Proposition 3.6 in [6] shows then that if α is a closed one-form one has then ε 2 1 2 α ε Tr (Jα). k∇H k ≥−4 H Therefore,

1 2 1 2 1 2 ∆ α ε ∆ ,εα, α ε Tr (Jα)+ Ric (α), α ε δ T (α), α + J (α), α ε. 2 Hk k − h H i ≥−4 H h H i − h H iV ε h i When ε is large enough, the right hand side of the above inequality is bounded from below by 2 cε α which concludes the proof. k kε In the remainder of the section, we fix ε> 0 so that (8) is true. Lemma 2.18. Let α be a smooth and closed one-form, then in L2,

lim et∆H,ε α = 0. t + → ∞ Proof. Let α be a smooth and closed one-form and consider the functional

t∆H,ε 2 φ(t)= e α εdµ. M k k Z One has t∆H,ε t∆H,ε φ′(t) = 2 ∆ ,εe α, e α εdµ. Mh H i Z Since, for every t 0, et∆H,ε α is a closed one-form (it will be proved in Lemma 4.9 that det∆H,ε α = ≥ Qtdα = 0 where Qt is a semigroup on two-forms), one deduces from (8) that

φ′(t) 2cεφ(t). ≤− This implies

t∆H,ε 2 2cεt 2 e α εdµ e− α εdµ t + 0. M k k ≤ M k k → → ∞ Z Z

We now have:

Lemma 2.19. Let α be a smooth closed one-form, then for every t 0, et∆H,ε α α is an exact ≥ − smooth form.

10 Proof. Let α be a smooth closed form. We have then: t t t∆H,ε d s∆H,ε s∆H,ε e α α = e αds = ∆ ,εe αds − 0 ds 0 H Z t Z t t s∆H,ε s∆H,ε s∆H = e ∆ ,εαds = e dδ ,εαds = d e δ ,εαds . H H H Z0 Z0 Z0  Therefore et∆H,ε α α is an exact smooth form. − We can now finally conclude the proof of Theorem 2.1.

Proof. Let α be a smooth closed one-form. We denote by [α] H1 (M) its de Rham cohomology ∈ dR class. From the previous lemma, for any t 0, one has [et∆H,ε α] = [α]. Since, in L2, et∆H,ε α 0, ≥ → and from Hodge theory α [α] is continuous in L2, one concludes [α] = 0. → Remark 2.20. One can prove as in Proposition 3.2 in [9], that for any fiberwise linear map Λ from the space of two-forms into the space of one-forms, and for any x M, we have ∈ 1 2 inf (∆ η ε)(x) (∆ , + Λ d)η(x), η(x) ε η, η(x) ε =1 2 Hk k − h H ∞ ◦ i k k   1 2 inf (∆ η ε)(x) ∆ ,2εη(x), η(x) ε ≤ η, η(x) ε =1 2 Hk k − h H i k k   1 2 1 2 = inf Tr (Jη )+ Ric (η), η ε δ T (η), η + J (η), η ε η, η(x) ε =1 −4 H h H i − h H iV εh i k k   As a consequence, assume that L is a Laplacian on forms which has the same symbol as ∗ −∇H∇H and that satisfies d∆ = Ld. Assume moreover that there exist ε > 0 and a constant cε > 0 such H that for every smooth-one form α,

1 2 2 ∆ α ε Lα, α ε cε α ε. (8) 2 Hk k − h i ≥ k k 2 Then, one must have Q(α), α g cε α . This means that Q is canonical and the optimal one to h i ≥ k kε consider when applying the Bochner’s method. In particular, Corollary 3.3.14 in [15] is a corollary (in our framework) of Theorem 2.1.

3 Sub-Laplacians on forms and transverse Weitzenb¨ock formulas

In the next Section 4, we will generalize the horizontal Bochner’s method developed in the previous section to all k-forms. The first step in doing so is to construct the relevant horizontal Laplacians. In this section we show how to construct such horizontal Laplacians. We propose here a very general construction in the setting of sub-Riemannian manifolds, generalizing [18], which shall later be applied to the special case of totally geodesic foliations.

3.1 Sub-Laplacians on sub-Riemannian manifolds We will first consider operators on forms on a sub-Riemannian manifold (M, , g ). We define the H H corresponding cometric g such that if v , . . . , vn is an orthonormal basis of x and α , α T M, ∗ 1 H 1 2 ∈ x∗ then H n α1, α2 g∗ = α1(vi)α2(vi). h i H i X=1 11 Notice that g∗ is degenerate along the subbundle of forms vanishing on . For any two-tensor 2H H ξ Γ∞(T M⊗ ), we define tr ξ( , )= ξ(g∗ ). ∈ H × × M H 2 Let be an arbitrary connection on with Hessian X,Y = X Y X Y for every X,Y ∇ ∇ ∇ ∇ −∇∇ ∈ Γ∞(T M). Consider the corresponding horizontal Laplacian

2 L := tr , , (9) H H ∇× × n therefore defined in a local horizontal frame X1, , Xn by L = i=1 Xi Xi X Xi . · · · H ∇ ∇ −∇∇ i Let Ω = Ω(M)= n+mΩk be the graded of smooth forms on M. We want to ⊕k=0 P define an operator ∆ on forms satisfying the following properties. H (I) For any function f Ω0, we have ∆ f = L f. ∈ H H (II) For any form α M, we have ∆ dα = d∆ α. ∈ H H (III) The operator is of “Weizenb¨ock-type”, meaning that,

∆ = L C, H H − where the operator C has order zero as a differential operator.

The case of Ω0 and Ω1, this was studied in [4, 6, 18]. In particular, from [18, Prop 2.4], we know the following result.

Proposition 3.1. Let T be the torsion of . Define a connection ˆ by ∇ ∇

ˆ X Y = X Y T (X,Y ). ∇ ∇ − 0 1 Then there exists an operator ∆ on Ω Ω satisfying (I), (II) to (III) if and only if ˆ g∗ = 0. H ⊕ ∇ H Using the terminology of [14], the connection ˆ is called the adjoint connection of (see the ∇ ∇ Appendix). It has torsion T and clearly is the adjoint of ˆ . The condition ˆ g = 0 is − ∇ ∇ ∇ ∗ equivalent to being preserved under parallel transport and that H H

Z X,Y g = ˆ Z X,Y g + X, ˆ Z Y g , h i H h∇ i H h ∇ i H for any Z Γ (T M) and X,Y Γ ( ). We will show that the result of Proposition 3.1 ∈ ∞ ∈ ∞ H generalizes to k-forms. A Weitzenb¨ock formula for the sub-Laplacian is essentially found in [14, Theorem 2.4.2]. We want to make a different presentation of this result, using a practical unified notation related to the Fermion calculus in the proof of Proposition 2.10. We will also include a proof, showing that one does not need to assume that the connection we are working with is of Le Jan-Watanabe type as in [14]. Let Ψ = n+m Ψ denote the space of all smooth sections of ⊕i,j=1 (i,j) n+m i j •T ∗M •T M = T ∗M T M. ∧ ⊗∧ ∧ ⊕∧ i,j M=0

For any element ν Ψ introduce the corresponding linear operator Cν : T M T M by the ∈ ∧• ∗ →∧• ∗ following rules:

12 (i) For any ν ,ν Ψ and f Ψ , 1 2 ∈ ∈ (0,0)

Cν1+ν2 = Cν1 + Cν2 , Cfν1 = fCν1 and C1 = id;

(ii) For a form α Ψ , i 0, we have Cα = α , the exterior product with α; ∈ (i,0) ≥ ∧ (iii) For a vector field X Ψ , we have CX = ιX , the contraction by X; ∈ (0,1)

(iv) For any vector field X Ψ(0,1) and multi-vector field χ Ψ(0,j), j 0, we have CX χ = CχCX ; ∈ ∈ ≥ ∧

(v) If α Ψ(i,0) and χ Ψ(0,j), i 0, j 0, then Cα χ = CαCχ. ∈ ∈ ≥ ≥ ⊗ In other words, if ν = fα αi X Yj, then 1 ∧···∧ ⊗ 1 ∧···∧

Cν (η)= fα αi (ιY ιX η). 1 ∧···∧ ∧ j · · · 1 Example 3.2. Let S Γ (End(T M)) be a vector bundle endomorphism, which we can consider ∈ ∞ ∗ as an element in Ψ(1,1). Then for one-forms α1, α2 ...,αi

CS(α α αi)= S(α ) α αi + α S(α ) αi 1 ∧ 2 ···∧ 1 ∧ 2 ∧···∧ 1 ∧ 2 ∧···∧ + + α α S(αi), · · · 1 ∧ 2 ∧···∧ Using the above notation notation, and denoting by Rˆ the curvature tensor of ˆ we have the ∇ following result.

Proposition 3.3. Let L be the horizontal Laplacian defined relative to a connection as in (9). H ∇ Assume that its adjoint connection ˆ satisfies ˆ g∗ = 0. Introduce the operator δ on forms by ∇ ∇ H H δ η = tr ( η)( , ). H − H ∇× × · Then the operator ∆ = dδ δ d, H − H − H satisfies (I)-(III). In fact, if we define Ric = Ric , + Ric , , Rici,j Ψ , by 1 1 2 2 ∈ (i,j)

Ric1,1(v)= tr Rˆ( , v) , (α β) Ric2,2(v, w) = 2 Rˆ(v, w)α, β g∗ , (10) − H × × ∧ h i H the operator ∆ satisfies the Weitzenb¨ock identity H

∆ = L CRic. H H − Before completing the proof, we emphasize the following properties of our creation/annihilation operators. Let be an arbitrary connection with torsion T . ∇ (a) For any X Ψ and ν Ψ , observe that ∈ (0,1) ∈ (i,j) i j X Cν = C X ν + Cν X , CX Cν = CιX ν + ( 1) − CνCX . ∇ ∇ ∇ −

k k+i j k (b) If ν Ψ , then Cν :Ω Ω and CνΩ = 0 for any k < i. ∈ (i,j) → −

13 (c) Let Z1, . . . , Zn+m be an arbitrary local frame of tangent bundle and let β1, . . . , βn+m be the corresponding coframe. Observe that we can then write the exterior differential as

n+m n+m n+m 1 d = CT + Cβ Z = βi βj ι + βi Z (11) i ∇ i 2 ∧ ∧ T (Zi,Zj ) ∧∇ i i i,j i X=1 X=1 X=1

This equality follows by observing that if we write d′ for the expression on the right hand side of (11), then d (α β) = (d α) β + ( 1)kα d β for any form β and k-form α. We also have ′ ∧ ′ ∧ − ∧ ′ d (fα)= df α + fd α. Hence, d = d if the operators agree on one-forms, and it is simple to ′ ∧ ′ ′ verify that indeed dα(X,Y )= X α(Y ) Y α(X)+ α(T (X,Y )). ∇ −∇ (d) Let g be a Riemannian metric with corresponding identification ♯ : • T M • T M and ∗ → ♭ : • T M • T M and define Ψ Ψ, ν ν as the vector bundle map determined by → ∗ → 7→ ∗ V V (α χ) = ♭χ ♯α. Then C = Cν∗ . ⊗V ∗ ⊗V ν∗ Proof of Proposition 3.3. Let T denote the torsion of . Relative to and for any X,Y Γ (T M) ∇ ∇ ∈ ∞ define operators 2 δ[X Y ]= CX Y , L[X Y ]= . ⊗ − ∇ ⊗ ∇Y,X We extend this to an arbitrary section of χ Γ (T M 2) by linearity and define ∆[χ]= dδ[χ] ∈ ∞ ⊗ − − δ[χ]d. Clearly, ∆[χ] commutes with d. Let Z ,...,Zn m be a local basis of T M with corresponding coframe β ,...,βn m. If x M 1 + 1 + ∈ is an arbitrary point, we may choose this basis such that such that Zi(x) = 0. Evaluating at the ∇ point x, we obtain

m m 2 ∆[X Y ]= L[X Y ] Cβi CX Y,Z + Cβi C Z X Y ⊗ ⊗ − ∇ i ∇ i ∇ i=1 i=1 m X m X 2 + Cβi CX Z ,Y + Cβi CX Z Y + CX C Y T + (CX CT + CT CX ) Y ∇ i ∇∇ i ∇ ∇ i i X=1 X=1 2 2 We will use the identities CX CT = CT CX + CT (X, ) and X,Y Y,X = CR(X,Y ) T (X,Y ) for − · ∇ −∇ −∇ the result. Note that in the last formula, R(X,Y ) denotes the curvature endomorphism acting on the cotangent bundle, i.e. the element in Ψ1,1 determined by

R(X,Y ) : (α, v) (R(X,Y )α)(v)= α(R(X,Y )v), α v T ∗M T M. 7→ − ⊗ ∈ ⊗ We compute

m m ∆[X Y ]= L[X Y ] Cβ CX C + Cβ CX ⊗ ⊗ − i R(Y,Zi) i ∇T (Y,Zi) i=1 i=1 m X n X

+ Cβi C Z X Y + Cβi CX Z Y C Y T CX Cδ[X Y ]T + CT (X, ) Y ∇ i ∇ ∇∇ i − ∇ − ⊗ · ∇ i=1 i=1 X Xm

= L[X Y ]+ CR(Y, )X Cβi CR(Y,Zi)CX + C(T (X, )+ X) Y ⊗ · − · ∇ ∇ i=1 m X

+ Cβi X Z Y +T (Y,Zi) CX Y T Cδ[X Y ]T . ⊗ ∇∇ i − ∧∇ − ⊗ i X=1

14 Introduce Ric[X Y ] = Ric , [X Y ] + Ric , [X Y ] defined by ⊗ 1 1 ⊗ 2 2 ⊗

Ric , [X Y ](v) = Rˆ(Y, v)X, 1 1 ⊗ − (12) Ric , [X Y ](v, w) = X Rˆ(v, w)Y.  2 2 ⊗ ∧ It then follows from (17) and (19) that

m

∆[X X]= L[X X] CRic[X X] + C ˆ X X + Cβi X ˆ X . ⊗ ⊗ − ⊗ ∇ ⊗ ∇ Zi ∇ i ∇ X=1

Define ∆[g∗ ]=∆ , L[g∗ ] = L and Ric[g∗ ] = Ric = Ric1,1 + Ric2,2. Assume that ˆ g∗ = 0. H H H H H ∇ H Then for every x, there is an orthonormal basis X ,...,Xn of the horizontal bundle so that 1 H ˆ Xi(x) = 0. The result follows. ∇ 3.2 Taming metrics

Let be a connection on (M, , g ) such that its adjoint connection ˆ satisfies ˆ g∗ = 0. In ∇ H H ∇ ∇ order to have an L2-inner product of forms and perform the Bochner’s method, we introduceH a Riemannian metric g such that g = g . Such a Riemannian metric is said to tame g . If |H H H we assume that this metric is compatible with our original connection , this turns out to be a ∇ surprisingly restrictive requirement.

Proposition 3.4 ([18]). Let be any connection with adjoint connection ˆ . Assume that ˆ g = ∇ ∇ ∇ ∗ 0. Then there exists a Riemannian metric g such that H

g = g , and g = 0, |H H ∇ if and only if ( X g)(Z,Z) = 0, and ( Z g)(X, X) = 0. (13) L L for any X Γ ( ) and Z Γ ( ). ∈ ∞ H ∈ ∞ H⊥ For the special case when := is integrable, the condition in (13) is equivalent to the V H⊥ foliation being Riemannian, bundle-like, with totally geodesic leaves. For more details, see [18]. F 4 Horizontal Bochner’s method on k-forms

In this Section, we now perform the horizontal Bochner’s method on k-forms, k 2, in the setting ≥ of Section 2, using the horizontal Laplacians constructed in the previous Section. So, the framework is the following. Let M be a smooth, oriented, connected, compact manifold with dimension n + m. We assume that M is equipped with a Riemannian foliation with bundle-like complete metric g F and totally geodesic m-dimensional leaves such that m 0, H ⊕ ε V and introduce the family of connections

ε 1 Y = X Y T (X,Y )+ JY X, 0 < ε + . ∇X ∇ − ε ≤ ∞

15 The corresponding horizontal Laplacian is then

∆ ,ε = dδ ,ε δ ,εd, H − H − H and it satisfies the Weitzenb¨ock identity given in Proposition 3.3, that is we have

ε ε ∆ ,ε = ( )∗ CRicε . H − ∇H ∇H − Remark 4.1. If α is a basic form, then ∆ , α is also basic and actually coincides with the so- H ∞ called basic Laplacian of the foliation. We refer to [24, 25] for a study of the basic Laplacian and of its connection to basic cohomology.

Remark 4.2. As in the one-form case, the operator CRicε is not symmetric in general.

4.1 Main result Our goal in this section is to prove the following result: Theorem 4.3. Let R be the curvature of the Bott connection. Define the horizontal curvature 2 2 operator R : ∗ ∗ as the linear operator determined by equation H ∧ H →∧ H R (β1 β2)(v, w)= R(♯Hβ1, ♯Hβ2)w, v , β1, β2 ∗, v, w . H ∧ h iH ∈H ∈H Assume that for some constant c> 0,

2 2 R (α), α c α , for any α ∗. (14) h H iH ≥ k kH ∈∧ H k Assume furthermore, that for any Z Γ ( ), we have Z T = 0. Then H (M) = 0 for every ∈ ∞ V ∇ dR m 0, a positive horizontal curvature operator implies that CRicε is strictly positive on forms α β where α is an i-form with i = 0,n vanishing on and an arbitrary ∧ 6 V form β vanishing on . In particular, it will be positive for k-forms with m 0 and has rank 1, we obtain that HdR( )=0for1 0, we obtain HdR( ) = 0 by Theorem 2.1, which also gives us HdR( ) = 0 by Poincar´eduality. As for the special case of oriented K-contact manifolds, R > 0 implies Q> 0 H since δ T = 0. H

16 Remark 4.5. Similarly to Remark 2.15, we can again compare this result with the result that can be obtained from Riemannian geometry of (M, , g). For simplicity, we discuss the case when the F leaves of have dimension m = 1. Let Z be a local unit length basis vector relative to g. Let α be F a two-form, define β = ιZ α and n 0 1 0 α = α ♭Z β = α ♭Xi ♭Xj. − ∧ 2 ij ∧ i,j X=1 gε If Rε is the curvature operator of R , then, after a long computation, we have that

0 0 Rεα, α ε = Rε(α + ♭Z β), α + ♭Z β ε h i h ∧ ∧ i 0 0 1 0 0 0 ε 2 = R (α ), α ε q(α , α ) ♭( J)Z ♯β, α g J ♯β, ♯β ε h H i − ε − h ∇ i − 4h i where n 0 0 1 0 0 q(α , α ) := α α T (Xi, Xj ), T (Xr, Xs) g 8 ij rsh i i,j,r,s X=1 n n 1 0 0 1 0 0 + α α T (Xi, Xr), T (Xj , Xs) g + α α T (Xi, Xs), T (Xr, Xs) g. 16 ij rsh i 16 ij rsh i i,j,r,s i,j,r,s X=1 X=1 Define

0 0 0 2 c = min R α , α g : α g = 1 > 0, k = min J X, X g : X g = 1 {h H 0 0i k 0 k } {−h i k k } 0 Mq = max q(α , α ) : α g = 1 < , M J = max ♭( J)Z X, α g : X = 1, α g = 1 < { k k } ∞ ∇ {h ∇ i k || k k } ∞ Then

0 0 Rε(α + ♭Z β), α + ♭Z β ε h ∧ ∧ i 0 2 1 0 2 M J 0 k 2 c α Mq α ∇ ♭Z β ε α ε + ♭Z β . ≥ k kε − ε k kε − √ε k ∧ k k k 4 k ∧ kε

Hence, as long as k > 0, we will have a positive lower bound of Rε for sufficiently large values of ε. However, at our level of generality, we may actually have k = 0, unless satisfies the property H that for every non-zero horizontal vector field X, T M is Lie generated by and [ , X] . H H 4.2 The horizontal Ricci curvature operator on forms

k (i,j) (i,j) For any k 0, we define an orthogonal decomposition T M = i j k T M, where T M ≥ ∧ ∗ ⊕ + = ∧ ∗ ∧ ∗ is spanned by all elements that can be written as α β where α and β are respectively an i-form ∧ vanishing on and a j-form vanishing on . Notice that if we define an inner product , ε with V H (i,j) h · · i respect to gε on covectors, and if α and β are sections of T M, then ∧ ∗ j α, β ε = ε α, β g. h i h i Define Ric = Ric∞. In other words, Ric = (Ric )1,1 +(Ric )2,2, where the terms are defined as in (10) withH respect to the curvature of theH Bott connectioH n. WeH have then the following result: Proposition 4.6. Assume that the horizontal curvature operator R has a positive lower bound H c as in (14). Then, there is a constant c > 0 such that for any α (i,j)T M, i = 0,n and any 1 ∈ ∧ ∗ 6 ε> 0, 2 C α, α ε c α . h RicH i ≥ 1k kε

17 The proof of this proposition relies on the following lemma:

Lemma 4.7. For k = 1, 2, let αk be an i-form vanishing on and let βk be a j-form vanishing on V . Then H

C (α β ), α β ε = C α , α ε β , β ε = α ,C α ε β , β ε. h RicH 1 ∧ 1 2 ∧ 2i h RicH 1 2i h 1 2i h 1 RicH 2i h 1 2i

Proof. We first observe that (Ric )1,1 vanishes on and has its image in by [17, Lemma 3.3 H V H (b)]. For (Ric )2,2, observe first that if X1, X2 Γ∞( ) and Z,W Γ∞( ), then H ∈ H ∈ V

R(Z,W )X , X g =  R(Z,W )X , X g =  ( Z T )(W, X ), X g = 0. h 1 2i h 1 2i h ∇ 1 2i If X , X ,W Γ( ) and Z Γ( ), we use the first Bianchi identity (18) and the relation (21), 1 2 ∈ H ∈ V Appendix A. Define A(X,Y ) := T (X,Y ) JX Y JY X. − − We then have relation

R(Z,W )X , X g h 1 2i 1 1 = ( Z A)(W, X ), X g ( W A)(Z, X ), X g 2h ∇ 1 2i − 2h ∇ 1 2i 1 1 ( X A)(X ,Z),W g + ( X A)(X ,Z),W g − 2h ∇ 1 2 i 2h ∇ 2 1 i 1 1 = Z,  ( W T )(X , X ) g = Z,  R∇(X , X )W g = 0. 2h ∇ 1 2 i 2h 1 2 i

If all vector fields are horizontal, we can again use (21) to verify that R(Z,W )X , X g = h 1 2i R(X1, X2)Z,W g. In conclusion, Ric∗ = Ric and CRicH (α β) = (CRicH α) β. h i H H ∧ ∧ We are now in position to prove Proposition 4.6.

Proof. By the result in Lemma 4.7, it is sufficient to consider α (i,0)T M. Hence, we can ignore ∈∧ ∗ the choice of ε, since all such elements have the same length, independent of metric. The remaining proof follows the lines of [16]. If X ,...,Xn is a local orthonormal basis of , define an operator 1 H R : T M T M ∧• ∗ →∧• ∗ n R = ♭Xj ιX ♭Xs ι − ∧ i ∧ R(Xi,Xj )Xi i,j,s X=1   n n = ♭Xj ι ♭Xj ♭Xs ι ιX − ∧ R(Xi,Xj )Xi − ∧ ∧ R(Xi,Xj )Xs i i,j i,j,s X=1 X=1 1 1 = C + C Cν = C Cν , (RicH)1,1 (RicH)2,2 − 2 0 RicH − 2 0 where n ν = ♭Xj ♭Xk Xi ( X T (Xj, Xk)) . 0 ∧ ⊗ ∧ ∇ i i,j,kX=1

18 n In the last expression, we have used that ♭Xj ♭Xs ι ιX = Cν, where − i,j,s=1 ∧ ∧ R(Xi,Xj )Xs i n P ν = ♭Xj ♭Xs Xi R(Xi, Xj)Xs − ∧ ⊗ ∧ i,j,s X=1 n 1 = ♭Xj ♭Xs Xi (R(Xi, Xj)Xs + R(Xj, Xs)Xi) −2 ∧ ⊗ ∧ i,j,s X=1 n (18) 1 = ♭Xj ♭Xs Xi R(Xj, Xs)Xi 2 ∧ ⊗ ∧ i,j,s X=1 n 1 ♭Xj ♭Xs Xi ( X T (Xj, Xs)) − 2 ∧ ⊗ ∧ ∇ i i,j,s X=1 1 = (Ric )2,2 ν0. H − 2

The operator Cν0 vanishes on horizontal forms, permitting us to write

C α, α ε = Rα, α . h RicH i h i Hence, it is sufficient to show that R is positive on (i,0)T M whenever i = 0,n. ∧ ∗ 6 Assume that the horizontal curvature operator is R is positive, and let S denote its square H root. Then we have the following relation

n R = R(♭Xi ♭Xj ), ♭Xr ♭Xs ♭Xj ιX (♭Xs ιX ) h ∧ ∧ i ∧ i ∧ r i,j,r,s X=1 n n 1 1 2 2 = ♭Xj ι♯ιv S(♭Xa ♭X ) ♭Xs ι♯ιX S(♭Xa ♭X ) = Sij = Sij −2 ∧ j ∧ b ∧ s ∧ b −2 − i,j i

P 2 R(α), α = Sij(α) . h i | | i

4.3 Ricci operator of ˆ ε under scaling ∇ ε ε If we consider now the operator ∆ ,ε and its Ricci operator Ric , determined by the curvature Rˆ ε H ε 1 of the connection ˆ , then the identity ˆ Y = X Y + JX Y , gives us ∇ ∇X ∇ ε

ε 1 1 Rˆ (X,Y )= R(X,Y )+ ( X J)Y ( Y J)X + J + [JX , JY ] ε ∇ − ∇ T (X,Y ) ε2 1 1 1 = R(X,Y )+ B (X,Y )+ B (X,Y )+ B (X,Y ), ε 1 ε 2 ε2 3

19 where B (X,Y ) = ( X J)Y ( Y J)X , B (X,Y )= J and B (X,Y ) = [JX , JY ]. Furthermore, 1 ∇ − ∇ 2 T (X,Y ) 3 for j = 1, 2, 3, we consider Bj as an element in Ψ(2,2) by formula

(α β)Bj(v, w)= Bj(v, w)♯Hα, ♯Hβ g, ∧ h i ˇ ˇ ˆ ˇ ˇ and define Bj in Ψ(1,1) by αBj(v) = tr Bj( , v)♯Hα, g. Notice that B3 = 0, B1 is the dual of 2 Hh × ×i δ T , while α(Bˇ2(v)) = J v, ♯Hα g. H h i With respect to the decomposition of the tangent space, we have that CB1 , CB2 and CB3 maps (i,j)T M into respectively (i 1,j+1)T M (i 2,j+2)T M, (i,j)T M and (i 2,j+2)T M, and ∧ ∗ ∧ − ∗ ⊕∧ − ∗ ∧ ∗ ∧ − ∗ identical relations hold for C for j = 1, 2, 3. We consider a k-form α as a sum α = α(i,j), Bˇj i+j=k α(i,j) (i,j)T M and use the convention that α(i,j) = 0 for i and j not in the permitted range ∈ ∧ ∗ P 0 i n and 0 j m. Then ≤ ≤ ≤ ≤

(C ε C )α, α ε (16) h Ric − RicH i 1 1 (i+1,j 1) (i,j) 1 (i+2,j 2) (i,j) = C ˆ α, α ε + C ˇ α − , α ε + CεB +B α − , α ε εh B2+B2 i ε h B1+B1 i ε2 h 1 3 i i+Xj=k i+Xj=k

We remark also that relative to local orthonormal bases X1,...,Xn and Z1,...,Zm, we have

CRicε CRicH −m n 1 1 = ♭δ T (Xi) ιXi + ♭Xk ♭( X T )(Xi, Xj) ιXj ιXi ε H ∧ ε ∧ ∇ k ∧ i X=1 i,j,kX=1 n m n 1 1 2 + ♭Zr ♭( Z T )(Xi, Xj ) ιX ιX + ♭J Xi ιX ε ∧ ∇ r ∧ j i ε ∧ i i,j r=1 i X=1 X X=1 n m n 1 1 + ♭Xi ♭Xj ιJ X ιX + ♭Zr ♭Zs ιJ J X ιX . 2ε ∧ ∧ T (Xj ,Xi) k k ε2 ∧ ∧ Zr Zs k k r,s=1 i,j,kX=1 X Xk=1 We have then the following lemma:

Lemma 4.8. Assume that Z T = 0 for any Z Γ∞( ). There exist a constant ε > 0 and a ∇ k ∈ V c2 2 constant c 0 such that for any α T M we have (C ε C )α, α ε α . 2 ≥ ∈∧ ∗ |h Ric − RicH i |≤ √ε k kε (i,j) Proof. Write α = i+j=k α according to the bi-grading. As we are assuming that the torsion (i+2,j 2) (i,j) is parallel in vertical directions, we obtain that CB α , α g = 0. Hence, by (16), if we P h 1 − i define Mj such that C ˇ α, β g Mj α g β g, then h Bj +Bj i ≤ k k k k

(C ε C )α, α ε h Ric − RicH i j 1 (i,j) 2 j 1 (i+1,j 1) (i,j) j 2 (i+2,j 2) (i,j) M ε − α + M ε − α − g α g + M ε − α − g α g ≤ 2 k kg 1 k k k k 3 k k k k i+Xj=k i+Xj=k i+Xj=k M2 (i,j) 2 M1 (i+1,j 1) (i,j) M3 (i+2,j 2) (i,j) α + α − ε α ε + α − ε α ε. ≤ ε k kε √ε k k k k ε k k k k i+Xj=k i+Xj=k i+Xj=k k Hence, there exists a constant c1 such that for any α T ∗M and for sufficiently large ε, we have c1 2 ∈∧ (C ε C )α, α ε α . h Ric − RicH i |≤ √ε k kε

20 4.4 Semigroup of ∆ ,ε H Similarly to the case for one-forms, we have a strongly continuous semigroup generated by ∆ ,ε. H Lemma 4.9.

(a) For 0 < ε , the operator ∆ ,ε is hypoelliptic. ≤∞ H (b) Let 0 <ε< . The operator ∆ ,ε is the generator of a strongly continuous and smoothing ∞ H 2 t∆H,ε semigroup of bounded operators on L ( •T ∗M, gε). Moreover, if e ,t 0 denotes this ∧ t∆ { ≥ } semigroup, then for every smooth one-form α, αt = e H,ε α is the unique solution of the heat equation:

∂αt =∆ ,εαt, ∂t H (α0 = α.

Furthermore, det∆H = et∆H d.

Proof.

(a) For 0 < ε , we can write ≤∞ ε ε ∆ ,ε = ( )∗ CRicε H − ∇H ∇H − ε ε where CRicε is a zero-order differential operator. Furthermore, we can locally write ( )∗ = n ε 2 ε ∇H ∇H ε i=1 Xi Xi with X1, , Xn being a local horizontal frame that satisfies the two- − ∇ −∇∇Xi · · · stepP H¨ormander’s  condition. Again one can deduce hypoellipticity from arguments similar to Proposition 3.5.1 in [27].

(b) The argument is identical to the one given for one-forms in Proposition 2.13.

4.4.1 Proof of Theorem 4.3 The proof of the main statement is now a direct consequence of the following lemma and otherwise identical to that of Theorem 2.1.

Lemma 4.10. Assume that R > 0 and that ZT = 0 for any Z Γ∞( ). There exists a value H ∇ ∈ V ε> 0 such that:

(a) There exists a constant c3 > 0 such that if α is a closed k-form with m

1 2 2 ∆ ,ε α ε ∆ ,εα, α ε c3 α ε. 2 H k k − h H i ≥ k k

(b) If α is a closed k-form with m

lim et∆H,ε α = 0. t + → ∞

(c) If α is a closed k-form with m

21 Proof.

(a) Notice that

1 2 ε 2 ∆ ,ε α ε ∆ ,εα, α ε = α ε + CRicε α, α ε CRicε α, α ε 2 H k k − h H i k∇H k h i ≥ h i

k Since we assumed that the horizontal curvature operator was positive and since T ∗M n 1 m (i,j) ⊆ − T M, we have from Lemma 4.8 and Proposition 4.6 that there are constants ⊕i=1 ⊕j=0 ∧ ∗ V c > 0 and c 0 such that 1 2 ≥ c2 ε ε 2 2 CRic α, α ε = CRicH α, α ε + CRic RicH α, α ε c1 α ε α ε. h i h i h − i ≥ k k − √εk k

The result follows.

(b) The proof is identical to Lemma 2.18.

(c) The proof is identical to Lemma 2.19.

A Appendix: Geometric identities about adjoint connections

In this Appendix we collect some formulas used in the text.

A.1 Adjoint connections For any connection on T M with torsion T , we define its adjoint connection ˆ by ∇ ∇

ˆ X Y = X Y T (X,Y ). ∇ ∇ − Notice that the curvature of ˆ is given by ∇

Rˆ(X,Y )Z = R(X,Y )Z ( X T )(Y,Z) ( Y T )(Z, X) − ∇ − ∇ + T (X, T (Y,Z)) + T (Y, T (Z, X)) T (T (X,Y ),Z) − = R(X,Y )Z d∇T (X,Y,Z) + ( Z T )(X,Y ) − ∇ = R(Z,Y )X R(Z, X)Y + ( Z T )(X,Y ). (17) − ∇ Here we have used the first Bianchi identity

 R(X,Y )Z = d∇T (X,Y,Z)= T (T (X,Y ),Z)+  ( X T )(Y,Z), (18) ∇ with  denoting the cyclic sum. Notice in particular that

Rˆ(X,Y )X = R(X,Y )X + ( X T )(X,Y ). (19) ∇

22 A.2 Connections compatible with the metric and commutation of the curvature Let be any connection compatible with the metric g and with torsion T . Introduce a map J by ∇ the formula JZ X,Y g = Z, T (X,Y ) g. Then it is simple to verify that h i h i g 1 X Y = Y + A(X,Y ), A(X,Y ) := T (X,Y ) JX Y JY X. ∇ ∇X 2 − −

Note that A(X,Y ),Z g = Y, A(X,Z) g. Computations yield the following. h i −h i Lemma A.1. For any connection , compatible with g, we have ∇

R∇(X,Y )Z,W g R∇(Z,W )X,Y g h i − h i 1 1 = ( X A)(Y,Z),W g ( Y A)(X,Z),W g 2h ∇ i − 2h ∇ i 1 1 ( Z A)(W, X),Y g + ( W A)(Z, X),Y g − 2h ∇ i 2h ∇ i 1 1 + A(T (X,Y ),Z),W g A(T (Z,W ), X),Y g 2h i − 2h i 1 1 + A(Y,Z), A(X,W ) g A(Z,Y ), A(W, X) g 4h i − 4h i 1 1 A(X,Z), A(Y,W ) g + A(Z, X), A(W, Y ) g − 4h i 4h i In particular,

(a) If T is skew-symmetric, i.e. T (X,Y ),Z g = Y, T (X,Z) g, then A(X,Y )= T (X,Y ) and h i −h i

R∇(X,Y )Z,W g R∇(Z,W )X,Y g (20) h i − h i 1 1 1 1 = ( X T )(Y,Z),W ( Y T )(X,Z),W g ( Z T )(W, X),Y + ( W T )(Z, X),Y g. 2h ∇ i− 2h ∇ i − 2h ∇ i 2h ∇ i

(b) If T (T (X,Y ),Z) = 0, then

R∇(X,Y )Z,W g R∇(Z,W )X,Y g (21) h i − h i 1 1 1 1 = ( X A)(Y,Z),W g ( Y A)(X,Z),W g ( Z A)(W, X),Y g + ( W A)(Z, X),Y g 2h ∇ i − 2h ∇ i − 2h ∇ i 2h ∇ i References

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