**HAL Id: hal-00927302**

**https://hal.archives-ouvertes.fr/hal-00927302**

Submitted on 14 Jan 2014

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire **HAL, est**
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

**A lower bound on the spectrum of the sublaplacian**

### Amine Aribi, Sorin Dragomir, Ahmad El Soufi

**To cite this version:**

Amine Aribi, Sorin Dragomir, Ahmad El Soufi. A lower bound on the spectrum of the sublaplacian.

Journal of Geometric Analysis, 2014, pp.1-28. �10.1007/s12220-014-9481-6�. �hal-00927302�

### Amine Aribi, Sorin Dragomir

^{1}

### , Ahmad El Soufi

^{2}

Abstract. We establish a new lower bound on the first nonzero eigen-
valueλ1(θ) of the sublaplacian∆*b* on a compact strictly pseudoconvex
CR manifold*M*carrying a contact formθwhose Tanaka-Webster con-
nection has pseudohermitian Ricci curvature bounded from below.

1. Introduction and statement of main result

By a classical result of A. Lichnerowicz (cf. Theorem D.I.1 in [8], p.

179) and M. Obata (cf. [34]) for any*m-dimensional compact Riemannian*
manifold (M,*g) with Ric** _{g}* ≥

*k g*the first nonzero eigenvalueλ

_{1}(g) of the Laplace-Beltrami operator∆

*satisfies the estimate*

^{g}(1) λ_{1}(g)≥*mk/(m*−1)

with equality if and only if*M*is isometric to the unit sphere*S** ^{m}*⊂R

*. The main ingredient in the proof of (1) is the Bochner-Lichnerowicz formula (cf.*

^{m+1}e.g. (G.IV.5) in [8], p. 131) (2) −1

2∆* ^{g}*
kduk

^{2}

= kHess(u)k^{2}−*g*

*Du*, *D*∆^{g}*u*

+Ric*g*(Du,*Du)*

for any*u* ∈*C*^{∞}(M,R). The great fascination exerted by the Lichnerowicz-
Obata theorem on the mathematical community in the last fifty years promp-
ted the many attempts to extend (2) and (1) to other geometric contexts e.g.

to Riemannian foliation theory (cf. S-D. Jung & K-R. Lee & K. Richard- son, [27], J. Lee & K. Richardson, [31], H-K. Pak & J-H. Park, [36]), to CR and pseudohermitian geometry (cf. E. Barletta & S. Dragomir, [3], E.

Barletta, [4], S-C. Chang & H-L. Chiu, [10], H-L. Chiu, [11], A. Green- leaf, [23], S. Ivanov & D. Vassilev, [24], S-Y. Li & H-S. Luk, [32]) and to sub-Riemannian geometry (cf. F. Baudoin & N. Garofalo, [7]). The present paper is devoted to a version of the estimate (1) occurring in CR geometry.

Given a compact strictly pseudoconvex CR manifold (M,*T*1,0(M)) endowed
with a positively oriented contact form θ, the pseudohermitian manifold

1Dipartimento di Matematica, Informatica ed Economia, Universit`a degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, Campus Macchia Romana, 85100 Potenza, Italy, sorin.dragomir@unibas.it

2Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e*Fran¸cois Rabelais,*
Tours, France,

amine.aribi@lmpt.univ-tours.fr,Ahmad.Elsoufi@lmpt.univ-tours.fr 1

(M, θ) carries a natural second order, positive, formally self-adjoint oper-
ator ∆*b* (the *sublaplacian* of (M, θ)), formally similar to the Laplacian in
Riemannian geometry, yet only degenerate elliptic (in the sense of J-M.

Bony, [9]). However∆*b*is hypoelliptic and (by a result of A. Menikoff& J.

Sj¨ostrand, [33]) it has a discrete spectrum

σ(∆*b*)= {λ_{ν}(θ) :ν ∈Z, ν≥0}, lim

ν→∞λ_{ν} = +∞,
λ_{0}(θ)=0, λ_{ν}(θ)≤ λ_{ν+1}(θ), ν ≥1.

On the other hand, by a result of N.Tanaka, [37], and S.M. Webster, [38],
(M, θ) carries a natural linear connection∇(the*Tanaka-Webster connection*
of (M, θ), cf. also [12], p. 25) whose Ricci tensor field is formally similar
to Ricci curvature in Riemannian geometry. It is then a natural problem to
look for a lower bound onλ_{1}(θ) whenever Ric∇is bounded from below. By
strict pseudoconvexity (M, θ) also carries a natural Riemannian metric *g*θ

(the*Webster metric*of (M, θ), cf. [12], p. 9) whose associated Riemannian
volume form is (up a multiplicative constant depending only on the CR
dimension*n)*Ψθ =θ∧(dθ)* ^{n}*. Let div :X(M)→

*C*

^{∞}(M,R) be the divergence operator associated to the volume formΨθ. Then the sublaplacian may be written in divergence form as∆

*b*

*u*= −div(∇

^{H}*u) where*∇

^{H}*u*(the

*horizontal*

*gradient) is the projection of the gradient*∇u with respect to

*g*

_{θ}, on the Levi, or maximally complex, distribution

*H(M)*= Re

*T*_{1,0}(M)⊕*T*_{0,1}(M).
Consequently the horizontal gradient∇^{H}*u*is the pseudohermitian analog to
the gradient*Du* in Riemannian geometry. The first step is then to produce
a pseudohermitian version of (2) i.e. compute∆* ^{b}*(k∇

^{H}*uk*

^{2}) (for an arbitrary eigenfunction

*u*of ∆

*b*) in terms of the pseudohermitian Hessian ∇

^{2}

*u*and the Ricci curvature Ric∇ of the Tanaka-Webster connection. The first to realize the difficulties in producing a pseudohermitian analog to (2) was A.

Greenleaf, [23]. Indeed his Bochner-Lichnerowicz type formula (3) 1

2∆*b*

k∇^{1,0}*uk*^{2}

= X

α,β

*u*_{αβ}*u*_{αβ}+*u*_{αβ}*u*_{αβ}

+2iX

α

(u_{α}*u*_{0α}−*u*_{α}*u*_{0α})+
+X

α,β

*R*_{αβ}*u*_{α}*u*_{β}+*in*X

α,β

*A*_{αβ}*u*_{α}*u*_{β}−*A*_{αβ}*u*_{α}*u*_{β}
+ 1

2 X

α

{u_{α}(∆*b**u)*_{α}+*u*_{α}(∆*b**u)*_{α}}
involves the torsion terms *A*_{αβ} (possessing no Riemannian counterpart).

Here ∇^{1,0}*u* = P

α*u*_{α}*T*_{α} (notations and conventions as used in (3) are ex-
plained in §2). However the attempt to confine oneself to the class of
Sasakian manifolds (M,*g*_{θ}) (as in [4], since Sasakian metrics*g*_{θ}have vanish-
ing pseudohermitian torsion i.e. *A*_{αβ} = 0) isn’t successful either: while tor-
sion terms may actually be controlled (when exploiting (3) integrated over
*M) by theL*^{2} norm of∇^{H}*u, the main technical di*fficulties really arise from
the occurrence of termsP

α(u_{α}*u*_{0α}−*u*_{α}*u*_{0α}) containing covariant derivatives

of ∇^{H}*u*in the ”bad” real direction *T* transverse to *H(M) (the* *Reeb vector*
field of (M, θ)).

The novelty brought by the present paper is to establish first a version
of Bochner-Lichnerowicz formula for a natural Lorentzian metric *F*_{θ} (the
*Fefferman metric*of (M, θ), cf. [29], [21]) on the total space of the canoni-
cal circle bundle*S*^{1} →*C(M)* −→^{π} *M. Fe*fferman metric*F*_{θ} was discovered
by C. Fefferman, [20], in connection with the study of boundary behav-
ior of the Bergman kernel of a strictly pseudoconvex domain in C* ^{n}*. An
array of problems of major interest in CR geometry e.g. the CR Yamabe
problem, [25], the study of subelliptic harmonic maps, [26], and Yang-
Mills fields on CR manifolds, [6], are closely tied to the geometry of the
Lorentzian manifold (C(M),

*F*

_{θ}). Indeed the aforementioned problems are projections on

*M*via π :

*C(M)*→

*M*of Lorentzian analogs to the corre- sponding Riemannian problems, as prompted by J.M. Lee’s discovery (cf.

[29]) thatπ∗ = ∆*b*, where is the Laplace-Beltrami operator of *F*_{θ} (the
wave operator on (C(M),*F*_{θ})). For instance any*S*^{1}-invariant harmonic map
Φ : (C(M),*F*_{θ}) → *N* into a Riemannian manifold*N* projects on a subellip-
tic harmonic mapφ: *M*→ *N*(in the sense of [26] and [6]). The arguments
in [8] carry over in a straightforward manner (cf. our §3) to Lorenzian
geometry and give (cf. (21) below)

(4) −1

2(F_{θ}(D f,*D f*))= *F*^{∗}_{θ}

*D*^{2}*f*, *D*^{2}*f*

−(D f)(*f*)+Ric* _{D}*(D f ,

*D f*) and the corresponding integral formula (22). The projection on

*M*of (4) then leads to another analog (similar to A. Greenleaf’s formula (3)) to Bochner-Lichnerowicz formula and then to a new lower bound on λ

_{1}(θ).

Precisely we may state

Theorem 1. *Let M be a compact, strictly pseudoconvex, CR manifold of*
*CR dimension n. Let* θ ∈ P_{+} *be a positively oriented contact form on M*
*and*∆*b**the corresponding sublaplacian. Let*Ric∇*be the Ricci tensor of the*
*Tanaka-Webster connection*∇*of*(M, θ)*and*λ_{1}(θ) ∈σ(∆*b*)*the first nonzero*
*eigenvalue of*∆^{b}*. If*

(5) Ric∇(X,*X)*≥*k G*_{θ}(X,*X)*

*for some constant k*> 0*and any X* ∈*H(M)then*
(6) λ_{1}(θ)≥ 2n

(n+2)(n+3) (

(n+3)k−(11n+19)τ_{0}− ρ_{0}
2(n+1)

)

*with*τ_{0} = sup* _{x∈M}*kτk

_{x}*and*ρ

_{0}=sup

*ρ(x)≥*

_{x∈M}*nk, where*τ

*and*ρ

*are respec-*

*tively the pseudohermitian torsion and scalar curvature of*(M, θ).

The lower bound (6) is nontrivial only for*k*sufficiently large (i.e. *k*must
satisfy (89) in§6). Let (*M,g*_{θ}) be a Sasakian manifold (equivalentlyτ= 0,

cf. e.g. [12]). Then under the same assumption (i.e. (5) in Theorem 1) A.

Greenleaf established the estimate (cf. [23])

(7) λ_{1}(θ)≥ *nk*

*n*+1.
Lower bound (6) is sharper that (7) when

(8) *k*> ρ_{0}

*n(n*+3).

If for instance *M* = *S*^{2n+1} is the standard sphere in C* ^{n+1}*, endowed with
the canonical contact form θ = (i/2)

∂−∂

|z|^{2}, thenρ_{0} = 2n(n+1) and
*k*= 2(n+1) hence (8) holds (and (6) is sharper than (7)).

The essentials of CR and pseudohermitian geometry are recalled in§2
(by following mainly [12]). The projection of (4) on*M*gives

(9) −1

2∆*b*

k∇^{H}*uk*^{2}

=

Π*H*∇^{2}*u*

2−(∇^{H}*u)(*∆*b**u)*+
+4(J∇^{H}*u)(u*_{0})− 3(n+1)

*n*+2 *A(∇*^{H}*u*, *J∇*^{H}*u)*+
+*n*+3

*n*+2Ric∇

∇^{H}*u*, ∇^{H}*u*

− ρ

2(n+1)(n+2)k∇^{H}*uk*^{2}

(the *pseudohermitian Bochner-Lichnerowicz* formula, cf. (79) in§5) and
the corresponding integral formula (80). The main technical difficulty in the
derivation of (9) is to compute the Ricci curvature Ric* _{D}*of the Lorentzian
manifold (C(M),

*F*

_{θ}). This is performed by relating the Levi-Civita con- nection

*D*of (C(M),

*F*

_{θ}) to the Tanaka-Webster connection∇of (M, θ) (cf.

(23)-(27) in §4, a result got in [6]) and adapting to*S*^{1} → *C(M)* → *M* a
technique originating in the theory of Riemannian submersions (cf. [35])
and shown to work in spite of the fact thatπ: (C(M),*F*_{θ})→ (M,*g*_{θ}) isn’t a
semi-Riemannian submersion (fibres ofπare degenerate). The relationship
among*D*and∇may then be exploited to compute the full curvature tensor
*R** ^{D}*. Only its trace Ric

*is evaluated in [29] and the formula there appears as too involved to be of practical use. Our result (cf. (50)-(54) in Lemma 3 below) is simple, elegant and local frame free. This springs from the decom- positions*

_{D}*T*(C(M))=Ker(σ)⊕R

*S*and Ker(σ)=

*H(M)*

^{↑}⊕R

*T*

^{↑}, themselves relying on the discovery (due to C.R. Graham, [21]) that σ ∈ Ω

^{1}(C(M)) (given by (20) below) is a connection 1-form in the principal circle bundle

*S*

^{1}→

*C(M)*→

*M. As a byproduct of Lemma 3 one reobtains the result by*J.M. Lee, [29], that none of the Fefferman metrics {F

_{θ}∈ Lor(C(M)) : θ ∈ P

_{+}}is Einstein. Integration of (9) over

*M*produces (by (78)) terms ku

_{0}k

_{L}^{2}where

*u*

_{0}≡

*T*(u) and

*u*is an arbitrary eigenfunction of ∆

*b*, corresponding to a fixed eigenvalueλ∈σ(∆

*b*). The

*L*

^{2}norm of the (restriction to the Levi distribution

*H(M) of the) pseudohermitian Hessian*Π

*H*∇

^{2}

*u*is estimated by

using (82) (a result got in [4]). Torsion terms and Ricci curvature terms are
respectively estimated by (87) and as a consequence of the assumption (5)
in Theorem 1 (together with (86)). Finally to control ku_{0}k* _{L}*2 one exploits
a fundamental result got in [10], and referred hereafter as the

*Chang-Chiu*

*inequality*(cf. (91) in Appendix A).

Spectral geometry (spectrae of Laplace-Beltrami and Schr¨odinger oper- ators) on compact Riemannian manifolds has been intensely investigated over the last twenty-five years by A. El Soufi & S. Ilias, [13], A. El Soufi

& S. Ilias & A. Ros, [14], A. El Soufi & B. Colbois, [15], A. El Soufi &

B. Colbois & E. Dryden, [16], A. El Soufi & N. Moukadem, [17], A. El Soufi & H. Giacomini & M. Jazar, [18], and A. El Soufi & R. Kiwan, [19].

A program aiming to recovering the quoted works in the realm of CR and pseudohermitian geometry was recently started by A. Aribi & A. El Soufi, [1]. As part of that program A. Aribi & S. Dragomir & A. El Soufi studied (cf. [2]) the dependence of spectrae of sublaplacians on the given contact form. The present work is another step on this path (studying spectrae of compact strictly pseudoconvex CR manifolds).

The paper is organized as follows. In§2 we recall the needed elements
of calculus on a pseudohermitian manifold (including the curvature theory
for the Tanaka-Webster conection, cf. [37], [38] and [12]). The Lorentzian
analog (4) to the Bochner-Lichnerowicz formula (2) is derived in§3. The
technicalities on curvature theory (needed to project (4) on*M) are dealt with*
in§4. In §5 we relate the Lorentzian Hessian*D*^{2}(u◦π) to the pseudoher-
mitian Hessian∇^{2}*u*and derive the pseudohermitian Bochner-Lichnerowicz
formula (9). The lower bound (1) is proved in§6. Appendix A contains a
proof of the Chang-Chiu inequality.

2. Areminder ofCRgeometry

For all definitions and basic conventions in CR and pseudohermitian ge-
ometry we rely on [12]. Let (M,*T*_{1,0}(M)) be an orientable CR manifold, of
CR dimension*n, and let*∂* _{b}* be the tangential Cauchy-Riemann operator. A
CR function is a

*C*

^{1}solution to the tangential C-R equations∂

_{b}*f*= 0. Let

*H(M) be the maximally complex, or Levi, distribution on*

*M*and let

*J*be its complex structure. LetPbe the space of all pseudohermitian structures on

*M. For each*θ ∈ P the Levi form is

*G*

_{θ}(X,

*Y)*= (dθ)(X,

*JY*) for every

*X,Y*∈

*H(M). If*

*M*is nondegenerate then each θ ∈ P is a contact form i.e. Ψθ = θ∧(dθ)

*is a volume form on*

^{n}*M. When*

*M*is strictly pseudo- convex we denote byP

_{+}the set of positively oriented contact forms i.e. all θ∈ Psuch that

*G*

_{θ}is positive definite. If

*M*is nondegenerate then for each contact formθ ∈ P there is a unique nowhere zero, globally defined, vec- tor field

*T*∈ X(M) (the Reeb vector field of (M, θ)) such that θ(T) = 1

and *T*⌋*dθ* = 0. By taking into account the direct sum decomposition
*T*(M)=*H(M)⊕*R*T* one may extend the Levi form*G*_{θ}to a semi-Riemannian
metric *g*_{θ} (the Webster metric of (M, θ)) given by *g*_{θ}(X,*Y*) = *G*_{θ}(X,*Y),*
*g*_{θ}(X,*T*) = 0 and*g*_{θ}(T,*T*) = 1, for every *X,Y* ∈ *H(M). By a fundamen-*
tal result of N. Tanaka, [37], and S. Webster, [38], for each contact form
θ ∈ P there is a unique linear connection ∇(the Tanaka-Webster connec-
tion of (M, θ)) such that i)*H(M) is parallel with respect to*∇, ii)∇g_{θ} = 0,

∇J = 0, and iii) the torsion tensor field *T*∇ is pure i.e. *T*∇(Z,*W)* = 0,
*T*∇(Z,*W)*=2iG_{θ}(Z,*W)T* andτ◦*J*+ *J*◦τ=0, for any*Z,W* ∈*T*1,0(M). For
all local calculations we consider a local frame{T_{α} : 1≤α≤*n}*of*T*_{1,0}(M),
defined on the open set*U, and set*

*g*_{αβ} =*G*_{θ}(T_{α}, *T*_{β}), *T*_{α} =*T*_{α}, ∇T* _{B}* = ω

_{B}

^{A}*T*

*, ω*

_{A}

_{B}*= Γ*

^{A}

^{A}*CB*θ

*, τ(T*

^{C}_{α})=

*A*

^{β}

_{α}

*T*

_{β},

*A*αβ =

*g*αγ

*A*

^{γ}

_{β},

α, β, γ,· · · ∈ {1,· · ·,*n},* *A,B,C,*· · · ∈ {0,1,· · ·,*n,* 1,· · ·,*n}.*

Here{θ^{α} : 1 ≤ α ≤ *n}* is the adpated coframe determined byθ^{α}(T_{β}) = δ^{α}_{β},
θ^{α}(T_{β})= 0 andθ^{α}(T)= 0. Then (cf. e.g. (1.62) and (1.64) in [12], p. 39-40)
(10) *dθ*=2ig_{αβ}θ^{α}∧θ^{β}, *dθ*^{α}= θ^{β}∧ω_{β}^{α}+θ∧τ^{α}, *A*_{αβ} =*A*_{βα},
where τ^{α} ≡ *A*^{α}

βθ^{β} and *A*^{α}

β = *A*^{α}_{β}. Therefore, if we set*A(X,Y)* = *g*_{θ}(τX,*Y)*
for any *X,Y* ∈ X(M) then *A* is symmetric. Let *R*^{∇} be the curvature tensor
field of the Tanaka-Webster connection ∇. As to the local components of
*R*^{∇} we adopt the convention*R*^{∇}(T*B*,*T**C*)T*A* = *R**A**D*

*BC**T**D* (cf. [12], p. 50).

The Ricci tensor of∇ is Ric∇(Y,*Z)* = tracen

*X*∈*T*(M)7−→*R*^{∇}(X,*Z)Y*o
for
any*Y*,*Z* ∈*T*(M). Locally we set*R** _{AB}*= Ric∇(T

*,*

_{A}*T*

*). The pseudohermitian Ricci tensor is then*

_{B}*R*

_{λµ}. By a result of S. Webster, [38] (to whom the notion is due)

*R*λµ =

*R*λα

αµ. The pseudohermitian scalar curvature isρ = *g*^{λµ}*R*λµ

whereh
*g*^{αβ}i

= h
*g*_{αβ}i−1

. Let us set Παβ

= *dω*_{α}^{β}−ω_{α}^{γ}∧ω_{γ}^{β}, Ωαβ

= Παβ−2iθ_{α}∧τ^{β}+2iτ_{α}∧θ^{β},
whereθ_{α} = *g*_{αβ}θ^{β},θ^{α} = θ^{α},τ_{α} = *g*_{αβ}τ^{β} andτ^{β} = *A*^{β}_{α}θ^{α}. By a result of S.M.

Webster, [38] (cf. also Theorem 1.7 in [12], p. 55)

(11) Ωαβ

=*R*_{α}^{β}_{λµ}θ^{λ}∧θ^{µ}+*W*_{αλ}^{β} θ^{λ}∧θ−*W*^{β}

αλθ^{λ}∧θ

where *W*_{αµ}^{β} = *g*^{βσ}∇_{α}*A*_{µσ} and *W*_{αλ}^{β} = *g*^{βσ}∇_{σ}*A*_{αλ}. Given*u* ∈ *C*^{∞}(M,R) the
pseudohermitian Hessian is (∇^{2}*u)(X,Y)* = (∇_{X}*du)Y* for any *X,Y* ∈ X(M).

Locally we set∇_{A}*u** _{B}* = (∇

^{2}

*u)(T*

*,*

_{A}*T*

*). The pseudohermitian Hessian is not symmetric. Rather one has the commutation formulae*

_{B}(12) ∇_{α}*u*_{β} =∇_{β}*u*_{α}, ∇_{α}*u*_{β}= ∇_{β}*u*_{α}−2ig_{αβ}*u*_{0}, *u*_{0} ≡ *T*(u),

(13) ∇_{0}*u*_{β} =∇_{β}*u*_{0}−*u*_{α}*A*^{α}_{β}.

The 3^{rd} order covariant derivative of *u*is (∇^{3}*u)(X,Y,Z)* = (∇_{X}*H** _{u}*)(Y,

*Z)*=

*X(H*

*(Y,*

_{u}*Z))*−

*H*

*(∇*

_{u}

_{X}*Y*,

*Z)*−

*H*

*(Y,∇*

_{u}

_{X}*Z) for anyX,Y*,

*Z*∈X(M), where

*H*

*≡*

_{u}∇^{2}*u. Locally we set* *u** _{ABC}* = (∇

^{3}

*u)(T*

*,*

_{A}*T*

*,*

_{B}*T*

*). Commutation formulae for*

_{C}*u*

*have been established by J.M. Lee, [30] (cf. also [12], p. 426) and are not needed through this paper. We shall use the divergence oper- ator div : X(M) →*

_{ABC}*C*

^{∞}(M,R) determined by L

*Ψθ = div(X)Ψθ for ev- ery*

_{X}*X*∈ X(M), where L

*is the Lie derivative. The horizontal gradient of*

_{X}*u*∈

*C*

^{1}(M,R) is ∇

^{H}*u*= Π

*H*∇u where Π

*H*:

*T*(M) →

*H(M) is the pro-*jection associated to the direct sum decomposition

*T*(M) =

*H(M)*⊕ R

*T*and ∇u is the ordinary semi-Riemannian gradient of

*u*with respect to

*g*

_{θ}i.e.

*g*θ(∇u,

*X)*=

*X(u) for any*

*X*∈ X(M). The sublaplacian of (M, θ) is the second order differential operator ∆

^{b}*u*= −div

∇^{H}*u*

, *u* ∈ *C*^{2}(M,R).

Another useful expression of the sublaplacian is∆*b**u* = −trace_{G}_{θ}Π*H*∇^{2}*u*or

∆*b**u*= −P2n
*a=1*

*E** _{a}*(E

*(u))−(∇*

_{a}

_{E}

_{a}*E*

*)(u) for any local*

_{a}*G*

_{θ}-orthonormal frame {E

*: 1 ≤*

_{a}*a*≤ 2n}of

*H(M) onU*⊂

*M. If*{T

_{α}: 1≤ α ≤

*n}*is a local frame of

*T*

_{1,0}(M) on

*U*⊂

*M*then

(14) ∆*b**u*= −∇_{α}*u*^{α}− ∇_{α}*u*^{α}.

A complex valued differential *p-form*ω ∈ Ω* ^{p}*(M)⊗C is a (p,0)-form (re-
spectively a (0,

*p)-form) if*

*T*

_{0,1}(M)⌋ω = 0 (respectively

*T*

_{0,1}(M)⌋ω = 0 and

*T*⌋ω = 0). Let Λ

*(M) →*

^{p,0}*M*andΛ

^{0,p}(M) →

*M*be the relevant bun- dles and Ω

*(M) and Ω*

^{p,0}^{0,p}(M) the corresponding spaces of sections. Let F be the flow on

*M*tangent to the Reeb vector

*T*(i.e.

*T*(F) = R

*T*). Let Ω

^{1,0}

*(F) = {ω ∈ Ω*

_{B}^{1,0}(M) :

*T*⌋ω = 0} be the space of all basic (1,0)- forms (on the foliated manifold (M,F), cf. also [5]). If ω ∈ Ω

^{1,0}

*(F) one may use the Levi form to define a unique complex vector field ω*

_{B}^{♯}∈

*C*

^{∞}(T

_{0,1}(M)). Here ω

^{♯}is determined by ω(Z) =

*G*

_{θ}(Z, ω

^{♯}) for any

*Z*∈

*T*

_{1,0}(M) hence locally ω

^{♯}= ω

^{β}

*T*

_{β}whereω

^{β}=

*g*

^{αβ}ω

_{α}andω = ω

_{α}θ

^{α}. Let δ

*: Ω*

_{b}^{1,0}

*(F) →*

_{B}*C*

^{∞}(M,C) be the differential operator (due to [30]) defined byδ

*ω =div*

_{b}ω^{♯}

andδ* _{b}*θ= 0 for anyω∈Ω

^{0,1}

*B*(F). Similarly ifη∈Ω

^{0,1}(M) then letη

^{♯}∈

*C*

^{∞}(T

_{1,0}(M)) be determined byη(Z)=

*G*

_{θ}(η

^{♯},

*Z),Z*∈

*T*

_{1,0}(M), and let us consider

δ* _{b}*:Ω

^{0,1}(M)→

*C*

^{∞}(M,C), δ

*η=div η*

_{b}^{♯}

, η∈Ω^{0,1}(M),

so that (locally) η^{♯} = η^{α}*T*_{α} where η = η_{β}θ^{β} and η^{α} = *g*^{αβ}η_{β}. Also δ* _{b}*ω =

∇_{β}ω^{β}andδ* _{b}*η=∇

_{α}η

^{α}. For each

*f*∈

*C*

^{∞}(M,C) we set (15) (P f)Z =

*g*

^{αβ}

∇^{3}*f*

(Z, *T*_{α}, *T*_{β})+2*n i A*

*Z,*(∇^{H}*f*)^{1,0}
,

(P f)Z =0, (P f)T = 0, *Z* ∈*T*_{1,0}(M).

Here*X*^{1,0} = Π^{1,0}*X*for any*X* ∈*H(M) and*Π^{1,0} :*H(M)*⊗C→ *T*1,0(M) is the
natural projection associated to*H(M)*⊗C = *T*1,0(M)⊕*T*0,1(M). Note that
*g*^{αβ}

∇_{T}_{β}(∇^{2}*f*)

(T_{α},*Z) is invariant under a transformationT*_{α}^{′} = *U*_{α}^{β}*T*_{β} with
deth

*U*_{α}^{β}i

, 0 on*U*∩*U*^{′}, hence (P f)Z is globally defined. Locally one has
*P f* =

*P*_{β}*f*

θ^{β}, *P*_{β}*f* = *f*_{β}^{α}_{α}+2ni A_{βγ}*f*^{γ},

(compare to Definition 1.1 and (1.2) in [10], p. 263). Similar to *P* :
*C*^{∞}(M,C)→Ω^{1,0}* _{B}* (F) we build

*P*:

*C*

^{∞}(M,C)→Ω

^{0,1}(M) given by

(16)

*P f*

*Z* =*g*^{αβ}

∇^{3}*f*

(Z, *T*_{β}, *T*_{α})−2*n i A*

*Z*, (∇^{H}*f*)^{0,1}
,
(P f)Z =0, (P f)T = 0, *Z* ∈*T*1,0(M),

where*X*^{0,1} = *X*^{1,0}for any*X* ∈*H(M). Also let*^{1}

(17) *P*_{0}*f* =δ* _{b}*(P f)+δ

*(P f),*

_{b}*f*∈

*C*

^{∞}(M,C).

From now on we assume that *M* is a compact strictly pseudoconvex CR
manifold and θ ∈ P_{+}. Then *g*_{θ} is a Riemannian metric on *M. It should*
be observed that the operators above are complexifications of real operators
familiar in Riemannian geometry, as follows. For instance let♯be ”raising
of indices” with respect to *g*θ i.e. *g*θ

α^{♯},*X*

= α(X) for any (real) 1-form
η ∈ Ω^{1}(M) and any (real) vector field *X* ∈ X(M). Then the musical iso-
morphisms ♯ : Ω^{1,0}*B* (F) → *C*^{∞}(T_{0,1}(M)) and ♯ : Ω^{0,1}(M) → *C*^{∞}(T_{1,0}(M))
(as built above) are restrictions of the C-linear extension (toΩ^{1}(M)⊗C =
*C*^{∞}(T^{∗}(M)⊗C)) of ♯ : Ω^{1}(M) → X(M) to Ω^{1,0}*B* (F) and Ω^{0,1}(M) respec-
tively. Also let Ω^{1}* _{B}*(F) be the space of all basic 1-forms on (

*M,*F) and

*d*

*:*

_{b}*C*

^{∞}(M) → Ω

^{1}

*B*(F) the first order differential operator given by

*d*

_{b}*u*=

*du*−

*u*0θ for every

*u*∈

*C*

^{∞}(M,R) where

*u*0 ≡

*T*(u). Let

*d*

^{∗}

*be the formal adjoint of*

_{b}*d*

*b*i.e.

*d*^{∗}* _{b}*ω ,

*u*

*L*^{2} = (ω , *d**b**u)** _{L}*2, ω ∈ Ω

^{1}

*B*(F),

*u*∈

*C*

^{∞}(M), with respect to the

*L*

^{2}inner products

(u,*v)*_{L}^{2} =
Z

*M*

*uv*Ψθ, (α, β)* _{L}*2 =
Z

*M*

*g*^{∗}_{θ}(α , β)Ψθ,

for any*u,v*∈*C*^{∞}(M,R) andα, β∈Ω^{1}(M). Let*d** _{b}*:

*C*

^{∞}(M,C)→Ω

^{1}

*(F)⊗C and*

_{B}*d*

_{b}^{∗}: Ω

^{1}

*B*(F)⊗C →

*C*

^{∞}(M,C) be the C-linear extensions of

*d*

*b*and

*d*

^{∗}

*. Then*

_{b}Lemma 1. i)Ω^{1}*B*(F)⊗C= Ω^{1,0}*B* (F)⊕Ω^{0,1}(M),ii)*d**b**f* = ∂_{b}*f* +∂_{b}*f for any*
*f* ∈*C*^{∞}(M,C),iii) *d*_{b}^{∗}

_{Ω}^{1,0}

*B* (F) =∂^{∗}* _{b}*= −δ

_{b}*,*iv)

*d*

^{∗}

_{b}_{Ω}0,1(M)= ∂^{∗}* _{b}* =−δ

_{b}*.*

1The operator*P*0in this paper and [10] differ by a multiplicative factor ^{1}_{4}.

Here the tangential C-R operator∂* _{b}*is thought of asΩ

^{0,1}(M)-valued (i.e.

one requests that *Z*⌋∂_{b}*f* = and*T*⌋∂_{b}*f* = 0 to start with). Also∂_{b}*f* is the
unique element of Ω^{1,0}*B* (F) coinciding with*d f* on *T*_{1,0}(M). Locally ∂_{b}*f* =
*f*_{α}θ^{α} and∂_{b}*f* = *f*_{α}θ^{α} where *f*_{α} ≡ *T*_{α}(*f*) and *f*_{α} ≡ *T*_{α}(*f*). Also ∂^{∗}* _{b}* and ∂

^{∗}

*are the formal adjoints of∂*

_{b}*:*

_{b}*C*

^{∞}(M,C) → Ω

^{1,0}

*(F) and∂*

_{B}*:*

_{b}*C*

^{∞}(M,C) → Ω

^{0,1}(M) with respect to the

*L*

^{2}inner products

(*f,g)** _{L}*2 =
Z

*M*

*f g*Ψ^{θ}, (ω_{1}, ω_{2})* _{L}*2 =
Z

*M*

*G*^{∗}_{θ}(ω_{1}, ω_{2})Ψ^{θ},

for any *f*,*g*∈*C*^{∞}(M,C) and any complex 1-formsω_{1}, ω_{2}either inΩ^{1,0}* _{B}* (F)
or in Ω

^{0,1}(M). Statements (i)-(ii) in Lemma 1 are immediate. The last equality in (iii) (respectively in (iv)) is due to [30] (cf. also [12], p. 280).

To prove (iii) one integrates by parts in (d^{∗}* _{b}*ω,

*f*)

_{L}^{2}. For every

*f*∈

*C*

^{∞}(M,R) Z

*M*

*g*^{∗}_{θ}

(P+*P)f* , *d*_{b}*f*

Ψθ =

*P f* +*P f* , *d*_{b}*f*

*L*^{2} =−(P_{0}*f* , *f*)* _{L}*2

(compare to (1.3) in [10], p. 263). By a result of S-C. Chang & H-L.

Chiu, [10], the operator*P*_{0}is nonnegative i.e. R

*M*(P_{0}*u)u*Ψθ ≥ 0 for any*u*∈
*C*^{∞}(M,R). We end the preparation of CR and pseudohermitian geometry by
stating the identity (a straightforward consequence of (14))

(18) *u*^{α} *u*αβ

β+*u*^{α} *u*αβ

β =−u^{α}*P*α*u*−*u*^{α}*P*α*u*+
+2ni

*A*_{αβ}*u*^{α}*u*^{β}−*A*_{αβ}*u*^{α}*u*^{β}

−

∇^{H}*u*
(∆*b**u).*

Compare to (2.3) in [10], p. 267.

3. Bochner-Lichnerowicz formulae onFefferman spaces

Let *S*^{1} → *C(M)* →^{π} *M* be the canonical circle bundle over a strictly
pseudoconvex CR manifold *M, of CR dimension* *n* (cf. e.g. Definition
2.9 in [12], p. 119). We set M = *C(M) for simplicity. Let* θ ∈ P_{+} be
a positively oriented contact form on *M* and let *F*_{θ} be the corresponding
Fefferman metric onMi.e.

(19) *F*_{θ} =π^{∗}*G*˜_{θ}+2 (π^{∗}θ)⊙σ,

(20) σ= 1

*n*+2
(

*dγ*+π^{∗} *i*ω_{α}^{α}− *i*

2*g*^{µν}*dg*_{µν}− ρ
4(n+1)θ

!) .

Cf. Definition 2.15 and Theorem 2.4 in [12], p. 128-129. As to the notations
in (19)-(20) we set ˜*G*_{θ} = *G*_{θ} on *H(M)*⊗*H(M) and ˜G*_{θ}(T,*W)* = 0 for every
*W* ∈ X(M). Moreover γ is a local fibre coordinate on M. We recall that
*F*_{θ} ∈ Lor(M) i.e. *F*_{θ} is a Lorentzian metric on M (a semi-Riemannian
metric of signature (−+· · ·+)).

Let *D* be the Levi-Civita connection of (M,*F*_{θ}). Given a point*z*_{0} ∈ M
let {E* _{p}* : 1 ≤

*p*≤ 2n + 2} be a local orthonormal (i.e.

*F*

_{θ}(E

*,*

_{p}*E*

*) = ǫ*

_{q}*δ*

_{p}*with ǫ*

_{pq}*∈ {±1}) frame of*

_{p}*T*(M), defined on an open neighborhood π

^{−1}(U) ⊂ M of

*z*

_{0}, such that (D

_{E}

_{p}*E*

*)(z*

_{q}_{0}) = 0 for any 1 ≤

*p,q*≤ 2n+ 2.

Such a local frame may always be built by parallel translating a given or-
thonormal basis {e* _{p}* : 1 ≤

*p*≤ 2n+2} ⊂

*T*

_{z}_{0}(M) along the geodesics of (M,

*F*

_{θ}) issuing at

*z*0. Let be the wave operator (the Laplace-Beltrami operator of (M,

*F*θ)). If

*f*∈

*C*

^{∞}(M,R) and

*g*=

*F*θ(D f,

*D f*) then

*g*=

−P2n+2
*p=1* ǫ* _{p}*n

*E**p*

*E**p*(g)

−
*D**E**p**E**p*

(g)o

. A calculation of (*g)(z*0), merely
adapting the proof of (G.IV.5) in [8], p. 131, to Lorentzian signature, leads
to

(21) −1

2(F_{θ}(D f,*D f*))= *F*^{∗}_{θ}

*D*^{2}*f*, *D*^{2}*f*

−(D f)(*f*)+Ric* _{D}*(D f,

*D f*).

Here Ric* _{D}*(X,

*Y*) = tracen

*Z*∈*T*(M)7→ *R** ^{D}*(Z,

*Y)X*o

and*R** ^{D}* is the curvature
tensor field of

*D. Let us assume thatM*is a closed manifold (i.e.

*M*is com- pact and∂M = ∅). ThenMis a closed manifold, as well (as the total space of a locally trivial bundle over a compact manifold, with compact fibres).

Integration of (21) over M leads (by Green’s lemma) to the (Lorentzian
analog to the)*L*^{2} Bochner-Lichnerowicz formula

(22) Z

M

n*F*^{∗}_{θ}

*D*^{2}*f* , *D*^{2}*f*

+Ric* _{D}*(D f,

*D f*)−(D f)(

*f*)o

*d*vol(F_{θ})= 0.

4. Curvature theory

By a result in [21] the 1-form σ ∈ Ω^{1}(M) is a connection form in the
canonical circle bundle *S*^{1} → M → *M. Let* *X*^{↑} ∈ X(M) denote the hor-
izontal lift of *X* ∈ X(M) i.e. *X*^{↑}* _{z}* ∈ Ker(d

*π) and (d*

_{z}*π)X*

_{z}

_{z}^{↑}=

*X*

_{π(z)}for any

*z*∈ M. Let

*S*∈ X(M) be the tangent to the

*S*

^{1}-action i.e. locally

*S*= [(n+2)/2]∂/∂γ. The Levi-Civita connection

*D*of (M,

*F*

_{θ}) is given by (cf. Lemma 2 in [6], p. 03504-26)

(23) *D*_{X}^{↑}*Y*^{↑}= (∇_{X}*Y)*^{↑}+{Ω(X,*Y*)◦π} *T*^{↑}+n
σh

*X*^{↑},*Y*^{↑}i

−2*A(X,Y)*◦πo
*S*,

(24) *D*_{X}^{↑}*T*^{↑}= {τ(X)+φ(X)}^{↑},

(25) *D*_{T}^{↑}*X*^{↑} =(∇_{T}*X*+φX)^{↑}+4 (dσ)(X^{↑},*T*^{↑})*S*,

(26) *D*_{X}^{↑}*S* = *D*_{S}*X*^{↑} = 1

2(JX)^{↑},

(27) *D*_{T}^{↑}*T*^{↑}= 2*V*^{↑}, *D*_{S}*S* = *D*_{S}*T*^{↑}= *D*_{T}^{↑}*S* =0,

whereΩ = −dθ whileφ : *H(M)* → *H(M) andV* ∈ *H(M) are the bundle*
endomorphism and vector field determined by

(28) *G*_{θ}(φX,*Y)*◦π= (dσ)(X^{↑}, *Y*^{↑}), *G*_{θ}(V,*X)*=(dσ)(T^{↑}, *X*^{↑}),
for any*X,Y* ∈ *H(M). Locally*φand*V* are given by

(29) φ_{α}^{β} = *i*
2(n+2)

(

*R*_{α}^{β}− ρ
2(n+1)δ^{β}_{α}

)

, φ_{α}^{β} = 0, φ_{α}^{0}= 0,

(30) *V*^{α} =*g*^{αβ}*V*_{β}, *V*_{β}= 1
2(n+2)

( 1

4(n+1)ρ_{β}+*i W*^{α}

αβ

) .

In particular [J, φ]=0 (as a consequence of (29)). We recall (cf. (1.100) in [12], p. 58)

(31) Ric*g*_{θ}(Tµ, *T*ν)= −1

2*R*µν+*g*µν,

(32) *R*_{µν} =*i(n*−1)*A*_{µν}, *R*_{0ν} = *S*^{µ}_{µν}, *R*_{µ0}= 0, *R*_{00} =0.

Here Ric_{g}_{θ} is the Ricci curvature of (M,*g*_{θ}). Also *S*(X,*Y)* = (∇* _{X}*τ)Y −
(∇

*τ)X for any*

_{Y}*X,Y*∈X(M), so that

*S*

^{µ}

_{µν}are among

*S*

_{kℓ}

^{j}*T*

*=*

_{j}*S*(T

*,*

_{k}*T*

_{ℓ}). As a consequence of (31) one has

*R*

_{µν}=

*R*

_{νµ}. Take the derivative of (20)

(n+2)*dσ*=π^{∗}
(

*idω*_{α}^{α}− *i*

2*dg*^{µν}∧*dg*µν− 1

4(n+1)*d(ρθ)*
)

and observe that*dg*^{µν}∧*dg*_{µν} =0. Also (by Theorem 1.7 in [12], p. 55)
*dω*_{α}^{α} = *R*_{µν}θ^{µ} ∧θ^{ν} +

*W*_{αλ}^{α} θ^{λ}−*W*_{αµ}^{α} θ^{µ}

∧θ . By (31)-(32)

(33) Ric∇(X,*JY)*=−2i

*R*_{µν}θ^{µ}∧θ^{ν}

(X,*Y)*−(n−1)A(X,*Y)*

for any*X,Y* ∈ *H(M). Alsod(ρθ)*=−ρΩon*H(M)*⊗*H(M). Consequently*
(34) 2(dσ)(X^{↑},*Y*^{↑})= 1

*n*+2
( ρ

2(n+1)Ω(X,*Y*)−

−(n−1)*A(X,Y)*−Ric∇(X,*JY*)}.

By a result in [28], Vol. I, p. 65, [X,*Y]*^{↑} is the horizontal component of
h*X*^{↑},*Y*^{↑}i

for any*X,Y* ∈ X(M). When *X,Y* ∈ *H(M) the vertical component*
may be easily derived from (34). One obtains the decomposition

(35) h

*X*^{↑}, *Y*^{↑}i

=[X,*Y]*^{↑}+ 2

*n*+2{Ric∇(X,*JY)*+
+(n−1)*A(X,Y)*− ρ

2(n+1)Ω(X,*Y)*
)

*S.*

Similarly let us compute *f* ∈ *C*^{∞}(M) in [X^{↑}, *T*^{↑}] = [X,*T*]^{↑} + *f S*. If ϕ =
*i*

*W*_{αλ}^{α}θ^{λ}−*W*_{αµ}^{α} θ^{µ}
then

*i*(dω_{α}^{α})(X,*T*)= (ϕ∧θ)(X,*T*)= 1
2ϕ(X),
2(n+2)(dσ)(X^{↑}, *T*^{↑})=ϕ(X)− 1

2(n+1)*d(ρθ)(X,T*)
or

(36) 2(dσ)(X^{↑},*T*^{↑})= 1
*n*+2

(

ϕ(X)− 1

4(n+1)*X(ρ)*
)

as*T*⌋*dθ*=0. We conclude (asσ(S)= ^{1}_{2})
(37) [X^{↑}, *T*^{↑}]= [X,*T*]^{↑}+ 2

*n*+2
( 1

4(n+1)*X(ρ)*−ϕ(X)
)

*S*.

Lemma 2. *Let M be a strictly pseudoconvex CR manifold, of CR dimension*
*n, and*θ ∈ P_{+} *a positively oriented contact form. The curvature R*^{D}*of the*
*Lorentzian manifold*(M,*F*_{θ})*is given by*

(38) *R** ^{D}*(X

^{↑},

*Y*

^{↑})Z

^{↑}=

*R*^{∇}(X,*Y*)Z^{↑}

−

− 1

2(n+1)(n+2) {X(ρ)Ω(Y,*Z)*−*Y(ρ)*Ω(X,*Z)}S*−

−*n*+5

*n*+2{(∇_{X}*A)(Y*,*Z)*−(∇_{Y}*A)(X,Z)}S*+
+ 1

*n*+2{(∇* _{X}*Ric∇)(Y,

*JZ)*−(∇

*Ric∇)(Y,*

_{Y}*JZ)}S*+ +Ω(Y,

*Z)*

(

(τX)^{↑}+(φX)^{↑}− ρ

4(n+1)(n+2)(JX)^{↑}
)

−

−Ω(X,*Z)*
(

(τY)^{↑}+(φY)^{↑}− ρ

4(n+1)(n+2)(JY)^{↑}
)

+ + 1

2(n+2) {Ric∇(Y,*JZ)*−(n+5)*A(Y*,*Z)}* (JX)^{↑}−

− 1

2(n+2) {Ric∇(X,*JZ)*−(n+5)*A(X,Z)}* (JY)^{↑}−

− 1
*n*+2

nRic∇(X,*JY) (JZ)*^{↑}−2Ω(X,*Y*) Ric∇(T,*JZ)S*o

−

− 1
*n*+2

(

(n−1)A(X,*Y*)− ρ

2(n+1)Ω(X,*Y)*
)

(JZ)^{↑}−

−2Ω(X,*Y)*
(

(φZ)^{↑}+ 2
*n*+2

"

ϕ(Z)− 1

4(n+1)*Z(ρ)*

#
*S*

) .

(39) *R** ^{D}*(X

^{↑},

*T*

^{↑})Z

^{↑}=

*R*^{∇}(X,*T*)Z^{↑}

+((∇* _{X}*φ)Z)

^{↑}+ + 1

*n*+2

nϕ(Z)(JX)^{↑}+ϕ(X)(JZ)^{↑}−

Ric∇(X,*JφZ)*+Ric∇(τX,*JZ)*
*S*o

−

− 1

4(n+1)(n+2)

n*Z(ρ)(JX)*^{↑}+*X(ρ)(JZ)*^{↑}o
+
+ 2

*n*+2
(

(∇* _{X}*ϕ)Z− 1

4(n+1)(∇_{X}*dρ)Z*
)

*S*−

− 1

*n*+2{(∇* _{T}*Ric∇)(X,

*JZ)*−(n+5)(∇

_{T}*A)(X,Z)}S*+ +{Ω(X, φZ)−Ω(τX,

*Z)}*

(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

−

−2Ω(X,*Z)*
(

*V*^{↑}− *T*(ρ)

4(n+1)(n+2)*S*
)

− 3(n+3)

*n*+2 {A(X, φZ)−*A(τX,Z)}S*,
(40) *R** ^{D}*(X

^{↑},

*S*)Z

^{↑}= − 1

2(n+2){Ric∇(X,*Z)*+(n+5)*A(X,JZ)}* *S*−

−1

2*G*θ(X,*Z)*
(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

,

(41) *R** ^{D}*(X

^{↑},

*Y*

^{↑})T

^{↑}=((∇

*τ)Y +(∇*

_{X}*φ)Y)*

_{X}^{↑}+4Ω(X,

*Y)V*

^{↑}−

− 1

*n*+2{Ric∇(JτX,*Y)*−Ric∇(X,*JτY)*+Ric∇(JφX,*Y)*−Ric∇(X,*JφY*)}*S*−

− *n*+5

2(n+2)^{2}{Ric∇(τX,*JY)*−Ric∇(JX, τY)+2(n−1)Ω(τX, τY)},
(42) *R** ^{D}*(X

^{↑},

*Y*

^{↑})S = 0,

*R*

*(T*

^{D}^{↑},

*S*)T

^{↑}=0,

*R*

*(T*

^{D}^{↑},

*S*)S =0,

(43) *R** ^{D}*(T

^{↑},

*S*)Z

^{↑}=

= 1
*n*+2

(

ϕ(JZ)−2ϕ(Z)− 1 4(n+1)

(JZ)(ρ)−2*Z(ρ)*
)

*S*,
*for any X,Y*,*Z* ∈*H(M).*

*Proof. AsH(M) is parallel with respect to*∇one has∇_{Y}*Z* ∈*H(M). Then*
(by (23) and (34))

(44) *D*_{X}^{↑}(∇_{Y}*Z)*^{↑} =(∇* _{X}*∇

_{Y}*Z)*

^{↑}+ Ω(X,∇

_{Y}*Z)*(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

+ + 1

*n*+2{Ric∇(X,*J∇*_{Y}*Z)*−(n+5)*A(X,*∇_{Y}*Z)}* *S*.
Next (by (23)-(24), (26), (34) and (44))

(45) *D*_{X}^{↑}*D*_{Y}^{↑}*Z*^{↑} =(∇* _{X}*∇

_{Y}*Z)*

^{↑}+

+{X(Ω(Y,*Z))*+ Ω(X,∇_{Y}*Z)}*

(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

+

− *X(ρ)*

2(n+1)(n+2)Ω(Y,*Z)S* − *n*+5

*n*+2 {X(A(Y,*Z))*+*A(X,*∇_{Y}*Z)}S*+
+ 1

*n*+2{X(Ric∇(Y,*JZ))*+Ric∇(X,*J∇*_{Y}*Z)}S*+
+Ω(Y,*Z)*

(

(τX)^{↑}+(φX)^{↑}− ρ

4(n+1)(n+2)(JX)^{↑}
)

+ + 1

2(n+2) {Ric∇(Y,*JZ)*−(n+5)*A(Y*,*Z)}* (JX)^{↑}.

The calculation of*D*_{[X}^{↑}_{,Y}^{↑}_{]}*Z*^{↑}is a bit trickier as [X,*Y]*<*H(M) in general. To*
start with one uses the decomposition (35) followed by [X,*Y]*= Π* ^{H}*[X,

*Y]*+ θ([X,

*Y])T*. This yields (by (26))

*D*_{[X}^{↑}_{,Y}^{↑}_{]}*Z*^{↑}= *D*_{[X,Y]}^{↑}*Z*^{↑}+ 2

*n*+2 *B(X,Y*)*D**S**Z*^{↑} =

= *D*_{(Π}_{H}_{[X,Y])}^{↑}*Z*^{↑}+θ([X,*Y])D*_{T}^{↑}*Z*^{↑}+ 1

*n*+2*B(X,Y) (JZ)*^{↑}
where we have set

*B(X,Y)*=Ric∇(X,*JY)*+(n−1)*A(X,Y*)− ρ

2(n+1)Ω(X,*Y)*

for simplicity. At this point we may use (23) (as Π*H*[X,*Y]* ∈ *H(M)) and*
(25) so that

*D*_{[X}^{↑}_{,Y}^{↑}_{]}*Z*^{↑}= ∇_{Π}_{H}_{[X,Y]}*Z*^{↑}

+ Ω(Π* ^{H}*[X,

*Y*],

*Z)T*

^{↑}−

−2n (dσ)

(Π*H*[X,*Y])*^{↑}, *Z*^{↑}

+*A*(Π*H*[X,*Y]*, *Z)*o
*S*+
+θ([X,*Y*])n

(∇_{T}*Z)*^{↑}+(φZ)^{↑}+4(dσ)(Z^{↑}, *T*^{↑})So
+ 1

*n*+2*B(X,Y) (JZ)*^{↑}.
Next (by*T* ⌋Ω =*T*⌋*A*=0 and the identities (34) and (36))

(46) *D*_{[X}^{↑}_{,Y}^{↑}_{]}*Z*^{↑}= ∇_{[X,Y]}*Z*^{↑}

+
+Ω([X,*Y],Z)*

(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

− *n*+5

*n*+2*A([X,Y*],*Z)S*+
+ 1

*n*+2

nRic∇(X,*JY) (JZ)*^{↑}+Ric∇(Π* ^{H}*[X,

*Y]*,

*JZ)S*o + + 1

*n*+2
(

(n−1)A(X,*Y*)− ρ

2(n+1)Ω(X,*Y)*
)

(JZ)^{↑}+
+θ([X,*Y*])

(

(φZ)^{↑}+ 2
*n*+2

"

ϕ(Z)− 1

4(n+1)*Z(ρ)*

#
*S*

) .

Moreover (by (45)-(46))
(47) *R** ^{D}*(X

^{↑},

*Y*

^{↑})Z

^{↑}=

[D_{X}^{↑}, *D*_{Y}^{↑}]−*D*_{[X}^{↑}_{,}_{Y}^{↑}_{]}

*Z*^{↑} =(∇* _{X}*∇

_{Y}*Z)*

^{↑}+ +{X(Ω(Y,

*Z))*+ Ω(X,∇

_{Y}*Z)}*

(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

−

− *X(ρ)*

2(n+1)(n+2)Ω(Y,*Z)S* − *n*+5

*n*+2 {X(A(Y,*Z))*+*A(X,*∇_{Y}*Z)}S*+
+ 1

*n*+2{X(Ric∇(Y,*JZ))*+Ric∇(X,*J∇*_{Y}*Z)}S*+
+Ω(Y,*Z)*

(

(τX)^{↑}+(φX)^{↑}− ρ

4(n+1)(n+2)(JX)^{↑}
)

+ + 1

2(n+2) {Ric∇(Y,*JZ)*−(n+5)*A(Y,Z)}* (JX)^{↑}−(∇* _{Y}*∇

_{X}*Z)*

^{↑}−

− {Y(Ω(X,*Z))*+ Ω(Y,∇_{X}*Z)}*

(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

+
+ *Y*(ρ)

2(n+1)(n+2)Ω(X,*Z)S* + *n*+5

*n*+2 {Y(A(X,*Z))*+*A(Y*,∇_{X}*Z)}S*−

− 1

*n*+2{Y(Ric∇(X,*JZ))*+Ric∇(Y,*J∇*_{X}*Z)}S*−

−Ω(X,*Z)*
(

(τY)^{↑}+(φY)^{↑}− ρ

4(n+1)(n+2)(JY)^{↑}
)

−

− 1

2(n+2) {Ric∇(X,*JZ)*−(n+5)*A(X,Z)}* (JY)^{↑}− ∇_{[X,Y]}*Z*^{↑}

−

−Ω([X,*Y],Z)*
(

*T*^{↑}− ρ

2(n+1)(n+2)*S*
)

+ *n*+5

*n*+2*A([X,Y*],*Z)S*−

− 1
*n*+2

nRic∇(X,*JY) (JZ)*^{↑}+Ric∇(Π*H*[X,*Y]*, *JZ)S*o

−

− 1
*n*+2

(

(n−1)A(X,*Y*)− ρ

2(n+1)Ω(X,*Y)*
)

(JZ)^{↑}−

−θ([X,*Y*])
(

(φZ)^{↑}+ 2
*n*+2

"

ϕ(Z)− 1

4(n+1)*Z(ρ)*

#
*S*

) . Using the identity

(48) [X,*Y]*= ∇_{X}*Y*− ∇_{Y}*X*+2Ω(X,*Y)T,* *X,Y* ∈*H(M),*
one has

*X(*Ω(Y,*Z))*+ Ω(X,∇_{Y}*X)*−*Y(*Ω(X,*Z))*−Ω(Y,∇_{X}*Z)*−Ω([X,*Y*],*Z)*=0
as∇Ω =0 and*T*⌋Ω =0. Similarly (again by (47) and*T*⌋*A*=0)

−X(A(Y,*Z))*−*A(X,*∇_{Y}*Z)*+*Y(A(X,Z))*+*A(Y*,∇_{X}*Z)*+*A([X,Y],Z)*=