Lichnerowicz Obata

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The Lichnerowicz-Obata theorem on sub-Riemannian

manifolds with transverse symmetries

Fabrice Baudoin, Bumsik Kim

Department of Mathematics, Purdue UniversityWest Lafayette, IN, USA

Abstract

We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type,we obtain a corresponding rigidity result: If the optimal lower bound for the firsteigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian sphere.

Contents

1 Introduction 1

2 The Bochner-Weitzenbock formula on sub-Riemannian manifolds with

transverse symmetries 3

3 Lichnerowicz estimate 5

4 The Obata sphere theorem onH-type manifolds 8

1 Introduction

The study of optimal lower bounds for sub-Laplacians on manifolds has attracted a lotof interest in the past few years. In particular, the most studied example has been theexample of the sub-Laplacian on CR manifolds. In that case, the story goes back at

least to the work by Greenleaf [12] which has seen, since then, several improvements andvariations. We mention in particular the works by Aribi-Dragomir-El Soufi [1], Barletta [2],Baudoin-Wang [7] and Li-Luk [21]. Some optimal lower bounds for the first eigenvalue ofsublaplacians also have been obtained in the context of quaternionic contact manifolds byIvanov-Petkov-Vassilev [16, 17]. More general situations were even considered by Hladky[13].

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In the present work, we obtain optimal first eigenvalue lower bounds in a large class of

sub-Riemannian manifolds that encompasses as a very special case Sasakian manifolds and3-Sasakian manifolds. This class is the class of sub-Riemannian manifolds with transversesymmetries that was introduced in [4]. Roughly speaking, a sub-Riemannian manifold withtransverse symmetries is a sub-Riemannian manifold for which the horizontal distributionadmits a canonical intrinsic complement which is generated by sub-Riemannian Killingfields. The lower bound we obtain in that case improves a previous lower bound that wasobtained by Baudoin-Kim in [5]. The method of [5] was to apply to an eigenfunction of thesub-Laplacian the curvature-dimension inequality proved in [4], and then to integrate thiscurvature-dimension inequality over the manifold. When used on a Riemannian manifold,this technique provides the optimal Lichnerowicz estimate. However, interestingly, thistechnique does not give the optimal estimate in the sub-Riemannian case and more work

is needed. Our approach here, is to take advantage of the Bochner-Weitzenbock formulathat was recently proved in [3] and to integrate this equality over the manifold. This givesan equality which when applied to an eigenfunction gives a better estimate than [5] forthe first eigenvalue. In the 1 or the 3-Sasakian case, the lower bound we obtain coincideswith the known optimal lower bound.

In the second part of the paper, we check the optimality of our lower bound, by provinga rigidity result in the spirit of Obata [22]. More precisely we prove the following result:

Theorem 1.1 LetM be a compact sub-Riemannian manifold ofH-type with dimensiond+h, d being the dimension of the horizontal bundle andh the dimension of the verticalbundle. Assume that for every smooth horizontal one-form,

RicH(), H 2H,

with >0, then the first eigenvalue1 of the sub-LaplacianL satisfies

1 d

d 1 + 3h.

Moreover, if 1 = dd1+3h , thenM is equivalent to a 1-Sasakian sphere S

2m+1(r) or a

3-Sasakian sphere S4m+3(r) for somer >0 andm 1.

This result generalizes to H-type manifolds, the corresponding result that was provedin Sasakian manifolds by Chang-Chiu [9] (see also [15, 20]) and in 3-Sasakian manifoldsby Ivanov-Petkov-Vassilev [18] . Like in the cited references, the main idea is to prove

that an extremal eigenfunction ffor the sub-Laplacian needs to satisfy 2f = f, forthe Levi-Civita connection of a well chosen Riemannian extension of the sub-Riemannianmetric.

The paper is organized as follows. Section 2 presents the basic materials on sub-Riemannianmanifolds with transverse symmetries. In particular, we present the Bochner-Weitzenbockformula that was proved in [3]. Section 3 is devoted to the proof of the lower bound forthe first eigenvalue and Section 4 proves its optimality in the context ofH-type manifolds.

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2 The Bochner-Weitzenbock formula on sub-Riemannian man-

ifolds with transverse symmetries

The notion of sub-Riemannian manifold with transverse symmetries was introduced in [4].We recall here the main geometric quantities and operators related to this structure and werefer to [3] and [4] for further details. We in particular focus on the Bochner-Weitzenbockformula that was proved in [3].

Let M be a smooth, connected manifold with dimension d+ h. We assume that M isequipped with a bracket generating distribution H of dimension d and a fiberwise innerproduct gH on that distribution. The distributionH is referred to as the set ofhorizontaldirections.

Definition 2.1 It is said thatM is a sub-Riemannian manifold with transverse symme-tries if there exists ah- dimensional Lie algebraVof sub-Riemannian Killing vector fieldssuch that for everyx M,

TxM =H(x) V(x).

We recall that a vector field Z is said to be a sub-Riemannian Killing vector field if theflow it generates, locally preserves the horizontal distribution and induces a gH-isometry.The distribution V is referred to as the set ofvertical directions. The choice of an innerproduct gV on the Lie algebra V naturally endows M with a one-parameter family ofRiemannian metrics that makes the decomposition H V orthogonal:

g = gH gV, > 0.

For notational convenience, we will often use the notation , instead ofg. The volumemeasure obtained as a product of the horizontal volume measure determined by gH andthe volume measure determined by gVwill be denoted by

The following connection was introduced in [4].

Proposition 2.2 (See [4]) There exists a unique connection onM satisfying the fol-lowing properties:

(i) g= 0, >0;

(ii) IfX andY are horizontal vector fields, XY is horizontal;

(iii) IfZ V, Z= 0;

(iv) IfX, Y are horizontal vector fields andZ V, the torsion vector fieldT(X, Y) isvertical andT(X, Z) = 0.

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Intuitively is the connection which coincides with the Levi-Civita connection of the

Riemannian metricg on the horizontal bundle H and that parallelizes the Lie algebra V.At every pointx M, we can find a local frame of vector fields {X1, , Xd, Z1, , Zh}such that on a neighborhood ofx:

(a) {X1, , Xd} is a gH-orthonormal basis ofH;

(b) {Z1, , Zh} is a gV-orthonormal basis of the Lie algebra V;

The sub-Laplacian on Mis the second-order differential operator which is given in a localadapted frame by

L=d

i=1

XiXi XiXi . (2.1)

We now introduce some tensors that will play an important role in the sequel. We defineRicH as the fiberwise symmetric horizontal linear map on one forms such that for everysmooth functionsf, g,

RicH(df), dgH= Ricci(Hf, Hg),

where Ricci is the Ricci curvature of the connection and H the horizontal gradient(projection of the gradient on the horizontal distribution).

For Z V, we consider the unique skew-symmetric map JZ defined on the horizontal

bundleH such that for every horizontal vector fields X and Y ,

gH(JZ(X), Y) =gV(Z, T(X, Y)). (2.2)

If (Zm)1mh is a basis of the Lie algebra V, the operatorh

m=1 JZm

JZm does not dependon the choice of the frame and shall concisely be denoted by JJ. We can note that inthe case where M is a Sasakian manifold, like the Heisenberg group for instance, JJ isthe identity map on the horizontal distribution.

IfVis a horizontal vector field, we consider then the smooth section on one-forms whichis given by in a local adapted frame by

TV= d

j=1

(T(V, Xj))j+ 1

2

hm=1

(JZmV)m.

It is easily seen that TVis a skew-symmetric operator for the Riemannian metric g2.

We finally recall the following definition that was introduced in [4]:

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Definition 2.3 The sub-Riemannian manifoldM is said to be of Yang-Mills type, if the

horizontal divergence of the torsion vanishes that is for every horizontal vector field X,and every adapted local frame

d=1

(XT)(X, X) = 0.

There are many interesting examples of Yang-Mills sub-Riemannian manifolds with trans-verse symmetries (see [4]). Sasakian and 3-Sasakian manifolds are examples of Yang-Millssub-Riemannian manifolds.

The following Bochner Weitzenbock formula was proved in [3].

Theorem 2.4 (Bochner-Weitzenbock formula [3]) Assume thatM is a sub-Riemannianmanifold with transverse symmetries of Yang-Mills type. For >0, we consider theg2-self-adjoint operator

= (H TH)

(H TH) +

1

2JJ RicH.

Then, for every smooth functionf onM,

dLf= df,

and for any smooth one-form,

1

2L22 , 2 = H T

H

22+

RicH()

1

2JJ(),

H

.

In a local adapted frame, we have

(H TH)

(H TH) =

di=1

(Xi TXi

)2 (XiXi TXiXi

),

and for any smooth one-form ,

H TH

2

2=

d

i=1

Xi TXi

2

2.

3 Lichnerowicz estimate

From now on, in all the paper we consider a compact Yang-Mills sub-Riemannian man-ifold M with transverse symmetries and adopt the notations of the previous section. InparticularLdenotes the sub-Laplacian on M. In this section, we prove the following result.

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Theorem 3.1 Assume that for every smooth horizontal one-form,

RicH(), H 12H, J

J(), H 2H,

and that for everyZ V,Tr(JZJZ) 2Z

2V,

with1, 2> 0 and 0. Then the first eigenvalue1 of the sub-LaplacianL satisfies

1 1

1 1d

+ 32

.

Before we prove the result, we briefly discuss the argument that was used in [5] to quicklyget, under the same assumptions, a lower bound on 1 which is less sharp.

Iff is a smooth function on M, from Theorem 2.4, we have

1

2Ldf22 dLf, df2 = Hdf T

Hdf

22+

RicH(df)

1

2JJ(df), df

H

.

Integrating this equality over Mand using the assumptions

RicH(), H 12H, J

J(), H 2H,

we deduce

M

dLf, df2

M

Hdf THdf

22+ 1

2M

df2H.

An integration by parts of left hand side of the inequality gives then

M

(Lf)2 2

M

dLf, dfV

M

Hdf THdf

22+

1

2

M

df2H. (3.3)

Now, a straightforward application of the Cauchy-Schwarz inequality yields the pointwiselower bound

Hdf THdf

2H

1

d(Lf)2 +

1

42df

2V. (3.4)

Coming back to (3.3), we infer then

d 1

d

M

(Lf)2 2

M

dLf, dfV

1

2

M

df2H+1

42

M

df2V.

In particular, ifLf=1f, then we obtain

d 1

d 21

M

f2 + 21

M

df2V

1

2

1

M

f2 +1

42

M

df2V.

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Choosing such that 21= 14

2 yields

1 1

1 1d

+ 42

.

This is not the optimal lower bound we are looking for. It is possible to improve this lowerbound from (3.3) by first integrating by parts the term

M

HdfTHdf

22 and, then

using Cauchy-Schwarz inequality. The key lemma is the following:

Lemma 3.2 ForfC(M),

M

Hdf THdf

22=

M

Hdf THdf

2H+ 2

M

Hdf3

2THdf

2

V

+12

M

Tr(JVfJVf) 52M

THdf2V.

Proof. Using the definition TH together with the Yang-Mills assumption, we see thatM

Hdf, TH(df)V=

1

4

M

Tr(JVfJVf). (3.5)

As a consequence, we obtainM

Hdf THdf

22

=M

Hdf THdf

2H+ 2

M

Hdf THdf

2V

=

M

Hdf THdf

2H+ 2

M

Hdf2V 4

M

Hdf, TH(df)V+ 2

M

THdf2V

(3.6)

By using (3.5), the trick is now to write

M

Hdf, TH(df)V=

3

2

M

Hdf, TH(df)V

1

2

M

Hdf, TH(df)V

=3

2

M

Hdf, TH(df)V

1

8

M

Tr(JVfJVf).

Coming back to (3.6) and completing the squares gives

M

Hdf THdf

22=

M

Hdf THdf

2H+ 2

M

Hdf 32THdf2

V

+1

2

M

Tr(JVfJVf) 5

2

M

THdf2V.

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We are now in position to complete the proof of Theorem 3.1.

Proof. Using the previous Lemma, Cauchy-Schwarz inequality and the assumptions

RicH(), H 12H, J

J(), H 2H, Tr(J

ZJZ) 2Z

2V

we get the lower boundM

Hdf THdf

22

1

d

M

(Lf)2 +3

42

M

df2V 5

8

M

df2H.

From (3.3), we know thatM

(Lf)2 2

M

dLf, dfV

M

Hdf THdf

22+

1

2

M

df2H. (3.7)

We thus deduce

d 1

d

M

(Lf)2 2

M

dLf, dfV

1

9

8

M

df2H+3

42

M

df2V.

If now f is such that Lf=1f, we get

d 1

d 21

M

f2 + 21

M

df2V

1

9

8

1

M

f2 +3

42

M

df2V.

Choosing such that

21=3

42,

gives the correct lower bound on 1.

4 The Obata sphere theorem on H-type manifolds

In this section we prove the optimality of the lower bound for the first eigenvalue of thesub-Laplacian on a special class of Yang-Mills manifolds by obtaining a rigidity result inthe spirit of the Obata sphere theorem.We first introduce the following definition inspired from the notion ofH-type groups thatwas introduced by Kaplan [19].

Definition 4.1 LetM be a sub-Riemannian manifold with transverse symmetries of Yang-Mills type. We will say thatM is ofH-type is for everyZ V, ZV= 1, the map JZ isorthogonal.

Sasakian or 3-Sasakian manifolds are examples of H-type manifolds. IfM is a H-typesub-Riemannian manifold, it is immediate from the definition that for Z, Z V,

JZJZ+ JZJZ=2Z, ZVIdH.

In particular, we haveJ2Z=Z

2VIdH.

In this section, we prove the following result:

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Theorem 4.2 LetMbe a compact sub-Riemannian manifold ofH-type. Assume that for

every smooth horizontal one-form,

RicH(), H 2H,

with >0, then the first eigenvalue1 of the sub-LaplacianL satisfies

1 d

d 1 + 3h.

Moreover, if 1 = dd1+3h

, thenM is equivalent to a 1-Sasakian sphere S2m+1(r) or a

3-Sasakian sphere S4m+3(r) for somer >0 andm 1.

To put things in perspective, we pause a little and describe the sub-Riemannian geometryof the 1 and 3 Sasakian spheres (see for instance [6, 8] for more details) and precise whatwe mean by equivalent in the previous theorem.

The sub-Riemannian geometry of the standard 1-Sasakian sphere S2m+1(1) is in-duced from the Riemannian structure of the complex projective space CPm by theHopf fibration S1 S2m+1 CPm. The sub-Laplacian L is then the lift of theLaplace-Beltrami operator on CPm. In that case, 1= 2m.

The sub-Riemannian geometry of the standard 3-Sasakian sphere S4m+3 is inducedfrom the Riemannian structure of the quaternionic projective space HPm by thequaternionic Hopf fibration SU(2) S4m+3 HPm. The sub-LaplacianL is then

the lift of the Laplace-Beltrami operator on HPn

. In that case, 1= m.

In the previous theorem, we use the following notion of equivalence for sub-Riemannianmanifolds with transverse symmetries: Two sub-Riemannian manifolds with transversesymmetries (M1, H1, V1) and (M2, H2, V2) are said to be equivalent if there exists anisomorphism M1 M2 that induces an isometry between the horizontal distributionsH1 and H2 and a Lie algebra isomorphism between V1 and V2.

We now discuss the cases that were already known in the literature. As we pointed outSasakian manifolds are ofH-type. In that case h= 1 and the lower bound becomes

1 d

d + 2

.

This estimate was obtained by Greenleaf [12] (see also [2]). The estimate is optimal andthe corresponding Obatas type rigidity result was obtained in [9] (see also [15] and [20]).The other case that was studied in the literature is the case of 3-Sasakian manifolds forwhichh = 3. The lower bound is then

1 d

d + 8.

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This bound was proved in [16, 17] and the corresponding rigidity result was obtained in

[18].

We now turn to the proof of Theorem 4.2. From now on, in the sequel, M will be a compactsub-Riemannian manifold ofH-type such that for every smooth horizontal one-form ,

RicH(), H 2H,

with >0. Since M is ofH-type, we have

JJ(), H= h2H,

and for every Z V,Tr(J

ZJZ

) =dZ2V

.

From Theorem 3.1, we get therefore the lower bound

1 d

d 1 + 3h.

The key lemma in our rigidity result is the following result:

Lemma 4.3 LetfC(M) such thatLf=1f with1= dd1+3h

. Thenf satisfies

2f(X, Y) =1d

fX, YH1

2T(X, Y)f, X, Y H. (4.8)

and

2f(X, Z) =21

2JZ(X)f, X H, Z V. (4.9)

Proof. From (3.7) we haveM

(Lf)2 2

M

dLf, dfV

M

Hdf THdf

22+

1

2

M

df2H,

and thus, since Lf=1f,

2

1M f

2

+ 21M df

2

VM Hdf T

Hdf

2

2+ 1

1

2

M f2

. (4.10)

On the other hand, from Lemma 3.2, we have

M

Hdf THdf

22

M

Hdf THdf

2H+ 2

M

Hdf 32THdf2

V

+2

2

M

df2V 5

8

M

df2H.

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It is readily checked that

Hdf THdf

2H=

2,#H f

2 +1

4Tr(JVfJVf),

where2,#H fdenotes the symmetrization of the horizontal Hessian off. Thus we have

M

Hdf THdf

22

M

2,#H f2 + 2

M

Hdf32THdf2

V

+32

4

M

df2V 5

81

M

f2.

Choosing such that 21= 324

and using the last inequality in (4.10) gives eventually

21 1

1

9

8

M

f2

M

2,#H f

2 + 2

M

Hdf3

2THdf

2

V

.

From Cauchy-Schwarz inequality, given the value of1, we always have21 1

1

9

8

M

f2

M

2,#H

f2

This means that, necessarily Hdf 32THdfV

= 0,

and moreover that2,#H f is a multiple ofgH. This immediately implies (4.8) and (4.9).

We are now in position to prove Theorem 4.2.Proof. Let f C(M) such that Lf = 1f with 1 =

dd1+3h . From the previous

lemma, we have

2f(X, Y) =1d

fX, YH1

2T(X, Y)f, X, Y H.

and

2f(X, Z) =21

d JZ(X)f, X H, Z V.

The trick is now that, since Mhas transverse symmetries, L commutes with any Z V

(see [4]), and thus Zf is also an eigenfunction for the same eigenvalue 1. In particularZfalso satisfies the equation (4.9). This gives for a horizontal vector field X andZ V,

3f(X,Z,Z) =421

d2 J2Z(X)f.

From the H-type assumption, we deduce

3f(X,Z,Z) =421d2

Z2VXf.

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Taking the trace and using the fact that both f and Zfare eigenfunctions ofL with

the same eigenvalue, we deduce that for any Z V,

Z2f=421d2

Z2Vf.

By polarization, it also implies that for every Z, Z V,

1

2(ZZ + ZZ)f=

421d2

Z, ZVf. (4.11)

Since M is compact, we easily see that V is a Lie algebra of compact type. We thereforecan choose forgVa biinvariant metric. We consider then the Riemannian metric on M,

g2 = gH 2gV,

where = 21d . By denoting the Levi-Civita connection associated to gR, it is then an

easy exercise to check that the previous relations imply then that for every smooth vectorfields X, Y

2f(X, Y) =1d

f g2(X, Y).

As a consequence of Obatas theorem [22], we deduce that (M, g2) is isometric to a sphere.Also by the very same Obatas theorem, the relations (4.11) imply that the Lie group Ggenerated by V is a sphere itself. This implies that this group is either S1 or SU(2).Morever, by the very definition of sub-Riemannian manifolds with transverse symmetries,G is seen to act properly on M. We deduce that there is a Riemannian submersion withtotally geodesic fibers

G M M/G.

The classification of Riemannian submersions with totally geodesic fibers of the spherethat was done in Escobales [11] completes our proof.

References

[1] A. Aribi, S. Dragomir & A. El Soufi, A lower bound on the spectrum of the sublapla-cian, To appear in Journal of Geom. Anal., (2014)

[2] E. Barletta, The Lichnerowicz theorem on CR manifolds. Tsukuba J. Math. 31(2007), no. 1, 77-97.

[3] F. Baudoin, Stochastic analysis on sub-Riemannian manifolds with transverse sym-metries, (2014), http://arxiv.org/abs/1402.4490

[4] F. Baudoin & N. Garofalo, Curvature-dimension inequalities and Ricci lowerbounds for sub-Riemannian manifolds with transverse symmetries, Arxiv preprint,http://arxiv.org/abs/1101.3590

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13/14

[5] F. Baudoin & B. Kim,Sobolev, Poincare and isoperimetric inequalities for subelliptic

diffusion operators satisfying a generalized curvature dimension inequality, To appearin Revista Matematica Iberoamericana (2014).

[6] F. Baudoin & J. Wang: The subelliptic heat kernel on the CR sphere. Math. Z. 275(2013), no. 1-2, 135-150.

[7] F. Baudoin & J. Wang: Curvature dimension inequalities and subelliptic heat kernelgradient bounds on contact manifolds, Potential Analysis, 40, 2014, 163-193.

[8] F. Baudoin & J. Wang:The subelliptic heat kernels of the quaternionic Hopf fibration,To appear in Potential Analysis, arXiv:1310.5938

[9] S-C. Chang & H-L. Chiu: Nonnegativity of CR Paneitz Operator and Its Application

to the CR Obatas Theorem, J Geom Anal (2009) 19: 261-287

[10] S. Dragomir & G. Tomassini, Differential geometry and analysis on CR manifolds,Birkhauser, Vol. 246, (2006).

[11] R. Escobales, Riemannian submersions with totally geodesic fibers. J. DifferentialGeom. 10 (1975), 253-276.

[12] A. Greenleaf,The first eigenvalue of a sublaplacian on a pseudohermitian manifold,Commun. Partial Differential Equations, 10 (1985), 191-217.

[13] R. Hladky, Bounds for the first eigenvalue of the horizontal Laplacian in positivelycurved sub-Riemannian manifolds. Geom. Dedicata 164 (2013), 155-177.

[14] R. Hladky,Isometries of complemented subRiemannian manifolds, (2013), To appearin Advances in Geometry, http://arxiv.org/abs/1203.1066

[15] S. Ivanov, A. Petkov & D. Vassilev,An Obata type result for the first eigenvalue of thesub-Laplacian on a CR manifold with a divergence-free torsion. J. Geom. 103 (2012),no. 3, 475-504.

[16] S. Ivanov, A. Petkov & D. Vassilev, The sharp lower bound of the first eigenvalue ofthe sub-Laplacian on a quaternionic contact manifold in dimension seven. NonlinearAnal. 93 (2013), 51-61.

[17] S. Ivanov, S., A. Petkov, & D. Vassilev,The sharp lower bound of the first eigenvalueof the sub-Laplacian on a quaternionic contact manifold, to appear in J. Geom. Anal.,arXiv:1112.0779.

[18] S. Ivanov, S., A. Petkov, & D. Vassilev,The Obata sphere theorems on a quaternioniccontact manifold of dimension bigger than seven, (2013), arXiv:1303.0409

[19] A Kaplan. Fundamental solutions for a class of hypoelliptic PDE generated by com-position of quadratic forms. Trans. Amer. Math. Soc., 258(1):147-153, 1980

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[20] S.Y Li & X. Wang,An Obata-type theorem in CR geometry. J. Differential Geom. 95

(2013), no. 3, 483-502.

[21] S.-Y. Li & H.-S. Luk, The sharp lower bound for the first positive eigenvalue of asub-Laplacian on a pseudohermitian manifold, Proc. Amer. Math. Soc. 132 (2004),no. 3, 789-798.

[22] M. Obata, Certain conditions for a Riemannian manifold to be isometric with asphere, J. Math. Soc. Japan 14 (1962), 333-340

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