THE OBATA SPHERE THEOREMS ON A QUATERNIONIC CONTACT MANIFOLD OF DIMENSION BIGGER THAN SEVEN S. IVANOV, A. PETKOV, AND D. VASSILEV

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1 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD OF DIENSION BIGGER THAN SEVEN S. IVANOV, A. PETKOV, AND D. VASSILEV Abstract. On a compact quaternionic contact (qc manifold of dimension bigger than seven and satisfying a Lichnerowicz type lower bound estimate we show that if the first positive eigenvalue of the sub-laplacian takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. The same conclusion is shown to hold on a non-compact qc manifold which is complete with respect to the associated Riemannian metric assuming the existence of a function with traceless horizontal Hessian. Contents 1. Introduction 1 2. Quaternionic contact manifolds Quaternionic contact structures and the Biquard connection Invariant decompositions The torsion tensor Torsion and curvature The Ricci identities, the divergence theorem Proof of the main Theorems Some basic identities Formulas for the derivatives of f The elliptic eigenvalue problem Formulas for the torsion tensors Formulas for the covariant derivatives of the torsion tensors Vanishing of the torsion The Riemannian Hessian Proof of Theorem Appendix The P form A new proof of Theorem References Introduction otivated by the classical Lichnerowicz 55 and Obata 62 theorems, earlier papers of the authors 36, 37 established a Lichnerowicz type lower bound estimate for the first eigenvalue of the sub-laplacian on a compact quaternionic contact (qc manifold. The case of equality in the lower bound estimate (Obatatype theorem was settled in the special case of a 3-Sasakian compact manifold where it was shown that the lower bound for the first eigenvalue of the sub-laplacian is achieved if and only if the 3-Sasakian manifold is isometric to the standard 3-Sasakian sphere. Quaternionic contact (qc structures were introduced by O. Date: August 28, athematics Subject Classification. 58C40,35P15,58J60. Key words and phrases. eigenvalue estimate, sub-laplacian, Obata sphere theorem, quaternionic contact. 1

2 2 S. IVANOV, A. PETKOV, AND D. VASSILEV Biquard 6 and are modeled on the conformal boundary at infinity of the quaternionic hyperbolic space. Thus, manifolds equipped with a qc structure are examples of sub-riemannian geometries. The (locally 3-Sasakian manifolds were characterized in 32, 40 by the vanishing of the torsion tensor of the Biquard connection. The qc geometry was a crucial geometric tool in finding the extremals and the best constant in the L 2 Folland-Stein Sobolev-type embedding, 23, 24, completely described on the quaternionic Heisenberg groups, 34, 35. In this paper we prove the full qc version of Obata s results for a general qc manifold of dimension bigger than seven. We find that the equality case of Lichnerowicz type inequality on a compact qc manifold of dimension at least eleven can be achieved only on the 3-Sasakian spheres. ore general, we show that on a complete with respect to the associated Riemannian metric qc manifold a certain (horizontal Hessian equation, cf. (1.6, allows a non-trivial solution if and only if the manifold is qc homothetic to the standard 3-Sasakian sphere. The qc seven dimensional case was considered in 37, however, the general qc Obata results in dimension seven remain open. Turning to some details, let us recall the mentioned classical results. Using the classical Bochner- Weitzenböck formula Lichnerowicz 55 showed that on a compact Riemannian manifold (, h of dimension n for which the Ricci curvature satisfies Ric(X, Y (n 1h(X, Y the first positive eigenvalue λ 1 of the (positive Laplace operator satisfies the inequality λ 1 n. Subsequently, Obata 62 proved that equality is achieved if and only if the Riemannian manifold is isometric to the round unit sphere. Obata observed that the trace-free part of the Riemannian Hessian of an eigenfunction f with eigenvalue λ = n vanishes, i.e., it satisfies the system (1.1 ( h 2 f = fh after which he defined an isometry using analysis based on the geodesics and Hessian comparison of the distance function from a point. In fact, Obata showed that on a complete Riemannian manifold (, h equation (1.1 allows a non-constant solution if and only if the manifold is isometric to the round unit sphere. In this case, the eigenfunctions corresponding to the first eigenvalue are the solutions of (1.1. Later, Gallot 26 generalized these results to statements involving the higher eigenvalues and corresponding eigenfunctions of the Laplace operator. The interest in relations between the spectrum of the Laplacian and geometric quantities justified the interest in Lichnerowicz-Obata type theorems in other geometric settings such as Riemannian foliations (and the eigenvalues of the basic Laplacian 51, 50, 47 and 63, to CR geometry (and the eigenvalues of the sub-laplacian 29, 4, 16, 14, 15, 17, 19, 52, and to general sub-riemannian geometries, see 5 and 31. In the CR case, Greenleaf 29 gave a version of Lichnerowicz result showing that if a compact strongly pseudo-convex CR manifold of dimension, n 3 satisfies a Lichnerowicz type inequality Ric(X, X 4A(X, JX (n 1g(X, X for all horizontal vectors X, where Ric and A are, correspondingly, the Ricci curvature and the Webster torsion of the Tanaka-Webster connection (in the notation from 44, 41, then the first positive eigenvalue λ 1 of the sub-laplacian satisfies the inequality λ 1 n. The standard (Sasakian CR structure on the sphere achieves equality in this inequality. Following 29 the above cited results on a compact CR manifold focused on adding a corresponding inequality for n = 1, 2 or characterizing the equality case mainly in the vanishing Webster-torsion case (the Sasakian case. The general case on a compact CR manifold satisfying the Lichnerowicz type condition was proved in 53, 54 using the results and the method of 42. This was achieved by introducing a new integration by parts step proving the vanishing of the Webster torsion assuming the first eigenvalue is equal to n (for the three dimensional case see 43. On the other hand, a generalization of the Obata result in the complete non-compact case was achieved in 42, where the standard Sasakian structure on the unit sphere was characterized through the existence of a non-trivial solution of a (horizontal Hessian equation on a complete with respect to the associated Riemannian metric CR manifold with a divergence free Webster torsion. To the best of our knowledge the case of a general torsion remains still open.

3 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 3 The main purpose of this paper is to prove the qc version of both results of Obata under no extra assumptions on the Biquard torsion when the dimension of the qc manifold is at least eleven, cf. Theorem 1.2 and Theorem 1.3. In particular, completeness rather than compactness is required in the second result, cf. Theorem 1.3, in contrast to the currently known CR case as mentioned in the previous paragraph. The quaternionic contact version of the Lichnerowicz result was found in 36 in dimensions grater than seven and in 37 in the seven dimensional case. The following result of 36 gives a lower bound on the positive eigenvalues of the sub-laplacian on a qc manifold. Theorem 1.1 (36. Let (, η, g, Q be a compact quaternionic contact manifold of dimension 3 > 7. Suppose that there is a positive constant k 0 such that the qc-ricci tensor and torsion of the Biquard connection satisfy the inequality (1.2 Ric(X, X 2( 5 T 0 (X, X 6(2n2 5n 1 (n 1( U(X, X k 0g(X, X, where Ric, T 0, U are, correspondingly, the Ricci curvature and the components of the torsion of the Biquard connection and X is a horizontal vector. Then, any eigenvalue λ of the sub-laplacian satisfies the inequality λ n n 2 k 0. The equality case of Theorem 1.1 is achieved on the 3-Sasakian sphere. It was shown in 35, see also 2, that the eigenspace of the first non-zero eigenvalue of the sub-laplacian on the unit 3-Sasakian sphere in Euclidean space is given by the restrictions to the sphere of all linear functions. The main results of this paper are the following two theorems. Theorem 1.2. Let (, η, g, Q be a compact quaternionic contact manifold of dimension 3 > 7 whose qc-ricci tensor and torsion of the Biquard connection satisfy the inequality (1.2. Then, the first positive eigenvalue λ of the sub-laplacian satisfies the equality (1.3 λ = n n 2 k 0 if and only if the qc manifold (, g, Q is qc-homothetic to the unit (3-dimensional 3-Sasakian sphere. According to 36, Remark 4.1, under the conditions of Theorem 1.1, an eigenfunction f corresponding to the first non-zero eigenvalue as in (1.3, f = n n2 k 0f, satisfies a linear PDE system, namely, the horizontal Hessian of f is given by (see Corollary 4.2 in the Appendix 1 (1.4 df(x, Y = 4(n 2 k 0fg(X, Y df(ξ s ω s (X, Y, where ξ 1, ξ 2, ξ 3 and ω 1, ω 2, ω 3 are the vertical Reeb vector fieds and the fundamental 2-forms, respectively. This brings us to our second main result, in which no compactness of is assumed a-priori, Theorem 1.3. Let (, η, g, Q be a quaternionic contact manifold of dimension 3 > 7 which is complete with respect to the associated Riemannian metric (1.5 h = g (η 1 2 (η 2 2 (η 3 2. Suppose there exists a non-constant smooth function f whose horizontal Hessian satisfies (1.6 df(x, Y = fg(x, Y df(ξ s ω s (X, Y. Then the qc manifold (, η, g, Q is qc homothetic to the unit (3-dimensional 3-Sasakian sphere. Clearly Theorem 1.3 implies Theorem 1.2 since any Riemannian metric on a compact manifold is complete and a qc-homothety allows us to reduce to the case k 0 = 4(n 2, which turns (1.4 in (1.6. We prove Theorem 1.3 by showing first that is isometric to the unit sphere S 3 and then that is qc-equivalent to the standard 3-Sasakian structure on S 3. To this effect we show that the torsion

4 4 S. IVANOV, A. PETKOV, AND D. VASSILEV of the Biquard connection vanishes and in this case the Riemannian Hessian satisfies (1.1 after which we invoke the classical Obata theorem showing that is isometric to the unit sphere. In order to prove the qc-equivalence part we show that the qc-conformal curvature vanishes, which gives the local qc-conformal equivalence with the 3-Sasakian sphere due to 39, Theorem 1.3. The existence of a global qc-conformal map between and the 3-Sasakian sphere follows, for example, from a qc Liouville-type result on the extension of a local (qc-conformal automorphism to a global one, see 10, Proposition for a general statement in the setting of Cartan geometries. In the Appendix, for completeness, we recall the notion of the P -function introduced in 37 and give a different proof of Theorem 1.1 based on the positivity of the P -function in the case n > 1 established in 37, Theorem 3.3. As a corollary of the proof, we show the validity of (1.4 for any eigenfunction of the sub-laplacian with eigenvalue given by (1.3. Convention 1.4. a We shall use X, Y, Z, U to denote horizontal vector fields, i.e. X, Y, Z, U H. b {e 1,..., e } denotes a local orthonormal basis of the horizontal space H. c The summation convention over repeated vectors from the basis {e 1,..., e } will be used. For example, for a (0,4-tensor P, the formula k = P (e b, e a, e a, e b means k = a,b=1 P (e b, e a, e a, e b. d The triple (i, j, k denotes any cyclic permutation of (1, 2, 3. e The sum (ijk means the cyclic sum. For example, df(i i Xω j (Y, Z = df(i 1 Xω 2 (Y, Z df(i 2 Xω 3 (Y, Z df(i 3 Xω 1 (Y, Z. (ijk e s will be any number from the set {1, 2, 3}, s {1, 2, 3}. Acknowledgments The research is partially supported by the Contract Idei, DID 02-39/ S.I and A.P. are partially supported by the Contract 168/2014 with the University of Sofia St.Kl.Ohridski. D.V. was partially supported by Simons Foundation grant # D.V. would like to thank Professor Luca Capogna for some useful comments. 2. Quaternionic contact manifolds In this section we will briefly review the basic notions of quaternionic contact geometry and recall some results from 6, 32 and 39 which we will use in this paper. It is well known that the sphere at infinity of a non-compact symmetric space of rank one carries a natural Carnot-Carathéodory structure, see 58, 60. In the real hyperbolic case one obtains the conformal class of the round metric on the sphere. In the remaining cases, each of the complex, quaternion and octonionic hyperbolic metrics on the unit ball induces a Carnot-Carathéodory structure on the unit sphere. This defines a conformal structure on a sub-bundle of the tangent bundle of co-dimension dim R K 1, where K = C, H, O. In the complex case the obtained geometry is the well studied standard CR structure on the unit sphere in complex space. Quaternionic contact (qc structure were introduced by O. Biquard, see 6, and are modeled on the conformal boundary at infinity of the quaternionic hyperbolic space. Biquard showed that the infinite dimensional family 49 of complete quaternionic-kähler deformations of the quaternion hyperbolic metric have conformal infinities which provide an infinite dimensional family of examples of qc structures. Conversely, according to 6 every real analytic qc structure on a manifold of dimension at least eleven is the conformal infinity of a unique quaternionic-kähler metric defined in a neighborhood of. Furthermore, 6 considered CR and qc structures as boundaries of infinity of Einstein metrics rather than only as boundaries at infinity of Kähler-Einstein and quaternionic-kähler metrics, respectively. In fact, in 6 it was shown that in each of the three cases (complex, quaternionic, octoninoic any small perturbation of the standard Carnot-Carathéodory structure on the boundary is the conformal infinity of an essentially unique Einstein metric on the unit ball, which is asymptotically symmetric. In the Riemannian case the corresponding question was posed in 22 and the perturbation result was proven in 28.

5 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 5 Another natural extension of an interesting Riemannian problem is the quaternionic contact Yamabe problem, a particular case of which 27, 65, 32, 34 amounts to finding the best constant in the L 2 Folland- Stein Sobolev-type embedding and the functions for which the equality is achieved, 23 and 24, with a complete solution on the quaternionic Heisenberg groups given in 34, Quaternionic contact structures and the Biquard connection. A quaternionic contact (qc manifold (, η, g, Q is a 3-dimensional manifold with a codimension three distribution H locally given as the kernel of a 1-form η = (η 1, η 2, η 3 with values in R 3. In addition H has an Sp(nSp(1 structure, that is, it is equipped with a Riemannian metric g and a rank-three bundle Q consisting of endomorphisms of H locally generated by three almost complex structures I 1, I 2, I 3 on H satisfying the identities of the imaginary unit quaternions, I 1 I 2 = I 2 I 1 = I 3, I 1 I 2 I 3 = id H which are hermitian compatible with the metric g(i s., I s. = g(.,. and the following compatibility condition holds 2g(I s X, Y = dη s (X, Y, X, Y H. The transformations preserving a given quaternionic contact structure η, i.e., η = µψη for a positive smooth function µ and an SO(3 matrix Ψ with smooth functions as entries are called quaternionic contact conformal (qc-conformal transformations. If the function µ is constant η is called qc-homothetic to η. The qc conformal curvature tensor W qc, introduced in 39, is the obstruction for a qc structure to be locally qc conformal to the standard 3-Sasakian structure on the ( 3-dimensional sphere 32, 39. A special phenomena, noted in 6, is that the contact form η determines the quaternionic structure and the metric on the horizontal distribution in a unique way. On a qc manifold with a fixed metric g on H there exists a canonical connection defined first by O. Biquard in 6 when the dimension ( 3 > 7, and in 21 for the 7-dimensional case. Biquard showed that there is a unique connection with torsion T and a unique supplementary subspace V to H in T, such that: (i preserves the decomposition H V and the Sp(nSp(1 structure on H, i.e. g = 0, σ Γ(Q for a section σ Γ(Q, and its torsion on H is given by T (X, Y = X, Y V ; (ii for ξ V, the endomorphism T (ξ,. H of H lies in (sp(n sp(1 gl(; (iii the connection on V is induced by the natural identification ϕ of V with the subspace sp(1 of the endomorphisms of H, i.e. ϕ = 0. This canonical connection is also known as the Biquard connection. When the dimension of is at least eleven 6 also described the supplementary distribution V, which is (locally generated by the so called Reeb vector fields {ξ 1, ξ 2, ξ 3 } determined by (2.1 η s (ξ k = δ sk, (ξ s dη s H = 0, (ξ s dη k H = (ξ k dη s H, where denotes the interior multiplication. If the dimension of is seven Duchemin shows in 21 that if we assume, in addition, the existence of Reeb vector fields as in (2.1, then the Biquard result holds. Henceforth, by a qc structure in dimension 7 we shall mean a qc structure satisfying (2.1. Notice that equations (2.1 are invariant under the natural SO(3 action. Using the triple of Reeb vector fields we extend the metric g on H to a metric h on T by requiring span{ξ 1, ξ 2, ξ 3 } = V H and h(ξ s, ξ k = δ sk. The Riemannian metric h as well as the Biquard connection do not depend on the action of SO(3 on V, but both change if η is multiplied by a conformal factor 32. Clearly, the Biquard connection preserves the Riemannian metric on T, h = 0. Since the Biquard connection is metric it is connected with the Levi-Civita connection h of the metric h by the general formula (2.2 h( A B, C = h( h AB, C 1 h(t (A, B, C h(t (B, C, A h(t (C, A, B, A, B, C Γ(T. 2 The covariant derivative of the qc structure with respect to the Biquard connection and the covariant derivative of the distribution V are given by (2.3 I i = α j I k α k I j, ξ i = α j ξ k α k ξ j. The vanishing of the sp(1-connection 1-forms on H implies the vanishing of the torsion endomorphism of the Biquard connection (see 32.

6 6 S. IVANOV, A. PETKOV, AND D. VASSILEV The fundamental 2-forms ω s of the quaternionic structure Q are defined by (2.4 2ω s H = dη s H, ξ ω s = 0, ξ V. Due to (2.4, the torsion restricted to H has the form (2.5 T (X, Y = X, Y V = 2ω 1 (X, Y ξ 1 2ω 2 (X, Y ξ 2 2ω 3 (X, Y ξ Invariant decompositions. An endomorphism Ψ of H can be decomposed with respect to the quaternionic structure (Q, g uniquely into four Sp(n-invariant parts Ψ = Ψ Ψ Ψ Ψ, where Ψ commutes with all three I i, Ψ commutes with I 1 and anti-commutes with the others two and etc. The two Sp(nSp(1-invariant components Ψ 3 = Ψ, Ψ 1 = Ψ Ψ Ψ are determined by Ψ = Ψ 3 3Ψ I 1 ΨI 1 I 2 ΨI 2 I 3 ΨI 3 = 0, Ψ = Ψ 1 Ψ I 1 ΨI 1 I 2 ΨI 2 I 3 ΨI 3 = 0. With a short calculation one sees that the Sp(nSp(1-invariant components are the projections on the eigenspaces of the Casimir operator Υ = I 1 I 1 I 2 I 2 I 3 I 3 corresponding, respectively, to the eigenvalues 3 and 1, see 11. If n = 1 then the space of symmetric endomorphisms commuting with all I s is 1-dimensional, i.e. the 3-component of any symmetric endomorphism Ψ on H is proportional to the identity, Ψ 3 = trψ 4 Id H. Note here that each of the three 2-forms ω s belongs to its -1-component, ω s = ω s1 and constitute a basis of the Lie algebra sp( The torsion tensor. The properties of the Biquard connection are encoded in the properties of the torsion endomorphism T ξ = T (ξ, : H H, ξ V. Decomposing the endomorphism T ξ (sp(n sp(1 into its symmetric part Tξ 0 and skew-symmetric part b ξ, T ξ = Tξ 0 b ξ, O. Biquard shows in 6 that the torsion T ξ is completely trace-free, tr T ξ = tr T ξ I s = 0, its symmetric part has the properties Tξ 0 i I i = I i Tξ 0 i I 2 (Tξ 0 2 = I 1 (Tξ 0 1, I 3 (Tξ 0 3 = I 2 (Tξ 0 2, I 1 (Tξ 0 1 = I 3 (Tξ 0 3, where the superscript means commuting with all three I i, indicates commuting with I 1 and anti-commuting with the other two and etc. The skew-symmetric part can be represented as b ξi = I i u, where u is a traceless symmetric (1,1-tensor on H which commutes with I 1, I 2, I 3. Therefore we have T ξi = Tξ 0 i I i u. If n = 1 then the tensor u vanishes identically, u = 0, and the torsion is a symmetric tensor, T ξ = Tξ 0. Any 3-Sasakian manifold has zero torsion endomorphism, T ξ = 0, and the converse is true if in addition the qc scalar curvature (see (2.6 is a positive constant 32 (the case of negative qc-scalar curvature can be treated very similarly, see 40, 41. We remind that a ( 3-dimensional Riemannian manifold (, g is called 3-Sasakian if the cone metric g c = t 2 h dt 2 on C = R is a hyper Kähler metric, namely, it has holonomy contained in Sp(n 1 9. A 3-Sasakian manifold of dimension ( 3 is Einstein with positive Riemannian scalar curvature ( 2( 3 48 and if complete it is a compact manifold with a finite fundamental group (see 8 for a nice overview of 3-Sasakian spaces Torsion and curvature. Let R =,, be the curvature tensor of and the dimension is 3. We denote the curvature tensor of type (0,4 and the torsion tensor of type (0,3 by the same letter, R(A, B, C, D := h(r(a, BC, D, T (A, B, C := h(t (A, B, C, A, B, C, D Γ(T. The qc-ricci tensor Ric, the normalized qc-scalar curvature S, the qc-ricci 2-forms ρ s, and the qc-ricci type-tensors ζ s are given by (2.6 Ric(A, B = R(e b, A, B, e b, 8n(n 2S = R(e b, e a, e a, e b, ρ s (A, B = 1 R(A, B, e a, I s e a, ζ s (A, B = 1 R(e a, A, B, I s e a. The sp(1-part of R is determined by the Ricci 2-forms and the connection 1-forms by (2.7 R(A, B, ξ i, ξ j = 2ρ k (A, B = (dα k α i α j (A, B, A, B Γ(T.

7 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 7 The two Sp(nSp(1-invariant trace-free symmetric 2-tensors T 0 (X, Y = g((t 0 ξ 1 I 1 T 0 ξ 2 I 2 T 0 ξ 3 I 3 X, Y, U(X, Y = g(ux, Y on H, introduced in 32, have the properties: (2.8 T 0 (X, Y T 0 (I 1 X, I 1 Y T 0 (I 2 X, I 2 Y T 0 (I 3 X, I 3 Y = 0, U(X, Y = U(I 1 X, I 1 Y = U(I 2 X, I 2 Y = U(I 3 X, I 3 Y. In dimension seven (n = 1, the tensor U vanishes identically, U = 0. We shall need the following identity taken from 39, Proposition 2.3 4T 0 (ξ s, I s X, Y = T 0 (X, Y T 0 (I s X, I s Y which implies the formula (2.9 T (ξ s, I s X, Y = T 0 (ξ s, I s X, Y g(i s ui s X, Y = 1 4 T 0 (X, Y T 0 (I s X, I s Y U(X, Y. We recall that a qc structure is said to be qc-einstein if the horizontal qc-ricci tensor is a scalar multiple of the metric, Ric(X, Y = 2(n 2Sg(X, Y. The horizontal Ricci-type tensor can be expressed in terms of the torsion of the Biquard connection 32 (see also 34, 39. We collect below the necessary facts from 32, Theorem 1.3, Theorem 3.12, Corollary 3.14, Proposition 4.3 and Proposition 4.4 with slight modification presented in 39 (2.10 Ric(X, Y = (2n 2T 0 (X, Y ( 10U(X, Y 2(n 2Sg(X, Y, ρ s (X, I s Y = 1 T 0 (X, Y T 0 (I s X, I s Y 2U(X, Y Sg(X, Y, 2 ζ s (X, I s Y = T 0 (X, Y 1 T 0 (I s X, I s Y 2n U(X, Y S g(x, Y, 2 T (ξ i, ξ j = Sξ k ξ i, ξ j H, S = h(t (ξ 1, ξ 2, ξ 3, g(t (ξ i, ξ j, X = ρ k (I i X, ξ i = ρ k (I j X, ξ j = h(ξ i, ξ j, X. For n = 1 the above formulas hold with U = 0. Hence, the qc-einstein condition is equivalent to the vanishing of the torsion endomorphism of the Biquard connection. In this case the normalized qc scalar curvature S is constant and the vertical distribution V is integrable 32 for n > 1 and 33 for n = 1. If S > 0 then the qc manifold is locally 3-Sasakian 32, (see 40 for the negative qc scalar curvature. We shall also need the general formula for the curvature 39, 41 (2.11 R(ξ i, X, Y, Z = ( X U(I i Y, Z ω j (X, Y ρ k (I i Z, ξ i ω k (X, Y ρ j (I i Z, ξ i 1 ( Y T 0 (I i Z, X ( Y T 0 (Z, I i X 1 ( Z T 0 (I i Y, X ( Z T 0 (Y, I i X 4 4 ω j (X, Zρ k (I i Y, ξ i ω k (X, Zρ j (I i Y, ξ i ω j (Y, Zρ k (I i X, ξ i ω k (Y, Zρ j (I i X, ξ i, where the Ricci two forms are given by, cf. 39, Theorem 3.1 or 41, Theorem (ρ s (ξ s, X = (X(S 1 2 ( e a T 0 (e a, X 3(I s e a, I s X 2( ea U(e a, X, (2.12 6(ρ i (ξ j, I k X = (2n 1(X(S 1 2 ( e a T 0 ( 1(e a, X 3(I i e a, I i X 4(n 1( ea U(e a, X The Ricci identities, the divergence theorem. We shall use repeatedly the following Ricci identities of order two and three, see also 39 and 36. Let ξ s be the Reeb vector fields, f a smooth function on

8 8 S. IVANOV, A. PETKOV, AND D. VASSILEV the qc manifold and f its horizontal gradient, g( f, X = df(x. We have: 2 f(x, Y 2 f(y, X = 2 ω s (X, Y df(ξ s, ( f(x, ξ s 2 f(ξ s, X = T (ξ s, X, f, 3 f(x, Y, Z 3 f(y, X, Z = R(X, Y, Z, f 2 ω s (X, Y 2 f(ξ s, Z, 3 f(x, Y, ξ i 3 f(y, X, ξ i = 2df(ξ j ρ k (X, Y 2df(ξ k ρ j (X, Y 2 3 f(ξ s, X, Y 3 f(x, ξ s, Y = R(ξ s, X, Y, f 2 f(t (ξ s, X, Y, ω s (X, Y 2 f(ξ s, ξ i, 3 f(ξ s, X, Y 3 f(x, Y, ξ s = 2 f (T (ξ s, X, Y 2 f (X, T (ξ s, Y df (( X T (ξ s, Y R(ξ s, X, Y, f. The sub-laplacian f and the norm of the horizontal gradient f of a smooth function f on are defined respectively by f = tr g H ( 2 f = df = 2 f(e a, e a, f 2 = df(e a df(e a. The function f is an eigenfunction with eigenvalue λ of the sub-laplacian if, for some constant λ we have (2.14 f = λf. From the Ricci identities we have the following formulas for the traces through the almost complex structures of the Hessian (2.15 g( 2 f, ω s = 2 f(e a, I s e a = df(ξ s. For a fixed local 1-form η and a fix s {1, 2, 3} the form V ol η = η 1 η 2 η 3 ωs 2n is a locally defined volume form. Note that V ol η is independent of s and the local one forms η 1, η 2, η 3 and therefore it is a globally defined volume form denoted with V ol η. The (horizontal divergence of a horizontal vector field/one-form σ Λ 1 (H defined by σ = tr H σ = σ(e a, e a supplies the integration by parts over compact formula 32, see also 65, (2.16 ( σ V ol η = Proof of the main Theorems The proof of Theorem 1.3 is lengthy and requires a number of steps which we present in the following subsections. Throughout this section we will work with the assumptions of Theorem 1.3. In particular, f is a non-constant smooth function whose horizontal Hessian satisfies (1.6. Our first step is to show the vanishing of the torsion tensor, T 0 = 0 and U = 0. We start by expressing the remaining parts of the Hessian (w.r.t. the Biquard connection in terms of the torsion tensors and show that f satisfies an elliptic equation on. A simple argument shows that T 0 (I s f, f = U(I s f, f = 0, s = 1, 2, 3. Furthermore, using the 1- component of the curvature tensor we show that T 0 (I s f, I t f = 0, s, t {1, 2, 3}, s t. In addition, we determine the torsion tensors T 0 and U in terms of the horizontal gradient of f and the tensor U( f, f. The analysis proceeds by finding formulas of the same type for the covariant derivatives of T 0 and U. Thus, the crux of the matter in showing that the torsion vanishes is the proof that U( f, f = 0. This fact will be achieved with the help of the Ricci identities, the contracted Bianchi second identity and thus far established results. In the next step of the proof of Theorem 1.3 we compute the Riemannian Hessian of f, with respect to the Levi-Civita connection of the metric (1.5 which allow us to invoke Obata s result thus proving that equipped with the Riemannian metric (1.5 is homothetic to the unit sphere in quaternion space. The final step is to show that is qc-homothetic to the ( 3-dimensional 3-Sasakian unit sphere. Here, we employ a standard monodromy argument showing that a compact simply connected locally qc-conformally flat manifold is globally qc-conformal to the 3-Sasakian unit sphere. For this we invoke the Liouville theorem

9 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 9 10, showing that every qc-conformal transformation between open subsets of the 3-Sasakian unit sphere is the restriction of a global qc-conformal transformation, i.e., an element of the group P Sp(n 1, Some basic identities. We start our analysis by finding a formula for the third covariant derivative of a function which satisfies (1.6. Lemma 3.1. With the assumptions of Theorem 1.3 we have the following formula for the third covariant derivative of the function f, (3.1 3 f(a, X, Y = df(ag(x, Y ω s (X, Y 2 f(a, ξ s, A Γ(T. Proof. The claimed formula is obtained by differentiating the Hessian equation (1.6. Indeed, the covariant derivative along A Γ(T of (1.6 gives 3 f(a, X, Y = df(ag(x, Y 2 f(a, ξ s ω s (X, Y df( A ξ s ω s (X, Y df(ξ s ( A ω s (X, Y, which together with (2.3 gives the identity, cf. also Convention 1.4 e, 3 f(a, X, Y = df(ag(x, Y (ijk 2 f(a, ξ i ω i (X, Y df( A ξ i ω i (X, Y df(ξ i ( A ω i (X, Y = df(ag(x, Y 2 f(a, ξ t ω t (X, Y t=1 α j (Adf(ξ k α k (Adf(ξ j ω i (X, Y α j (Aω k (X, Y α k (Aω j (X, Y df(ξ i (ijk (ijk = df(ag(x, Y 2 f(a, ξ t ω t (X, Y, t=1 which completes the proof. After this technical Lemma, our first goal is to find a formula for the curvature tensor R(Z, X, Y, f, for f satisfying (1.6, using Lemma 3.1 with A = Z, the Ricci identities (2.13, and the properties of the torsion. In fact, after some standard calculations it follows (3.2 R(Z, X, Y, f = df(zg(x, Y df(xg(z, Y df(ξ s, Zω s (X, Y df(ξ s, Xω s (Z, Y 2 df(ξ s, Y ω s (Z, X T (ξ s, Z, fω s (X, Y T (ξ s, X, fω s (Z, Y. By taking traces in (3.2 we can derive formulas for the various contracted tensors (2.6. We shall use the following, (3.3 Ric(Z, f = ( 1df(Z T (ξ s, I s Z, f 3 df(ξ s, I s Z, ζ i (I i Z, f = df(z ( 1T (ξ i, I i Z, f T (ξ j, I j Z, f T (ξ k, I k Z, f ( 1 df(ξ i, I i Z df(ξ j, I j Z df(ξ k, I k Z.

10 10 S. IVANOV, A. PETKOV, AND D. VASSILEV The above formulas imply some other basic identities to which we turn next. Note that with the help of (2.10 we can rewrite the Lichnerowicz type assumption (1.2 in the form (3.4 L(X, X def = 2(n 2Sg(X, X α nt 0 (X, X β nu(x, X k 0 g(x, X, X H, α n = 2(2n 3(n 2, β 4(2n 1(n 22 n = ((n 1, which allows to write the first claim of the following Lemma in the form L(Z, f = 0 for all Z H whenever f satisfies (1.6 taking k 0 = 4(n 2. Lemma 3.2. With the assumptions of Theorem 1.3, the next identity holds true (3.5 (S 2df(Z 2n 3 T 0 (Z, f Furthermore, we have 2(2n 1(n 2 U(Z, f = 0. ((n 1 (3.6 T 0 (I s f, f = 0, U(I s f, f = 0. Proof. The first equations in (3.3 and (2.10 together with (2.9 imply (3.7 3 df(ξ s, I s Z = 1 (2n 4S df(z (2n 3T 0 (Z, f ( 7U(Z, f. The sum over 1, 2, 3 of the second equality in (3.3 together with the third equality of (2.10 and (2.9 gives (3.8 ( 1 df(ξ s, I s Z = (3 6nSdf(Z (2n 3T 0 (Z, f 3U(Z, f. Subtracting (3.7 from (3.8 we obtain which for n > 1 yields (3.9 4(n 1 df(ξ s, I s Z = 4(1 n(1 Sdf(Z 4(n 1U(Z, f, The sum of (3.7 and (3.8 gives (3.10 ( df(ξ s, I s Z = (1 Sdf(Z n 1 U(Z, f. n 1 df(ξ s, I s Z = ((1 2Sdf(Z (2n 3T 0 (Z, f (2n 5U(Z, f. Equalities (3.9 and (3.10 imply (3.5. Letting Z = I s f in the latter it follows T 0 (I s f, f = 0 since U(I s f, f = Formulas for the derivatives of f. By assumption, the second order horizontal derivatives of f satisfy the Hessian equation (1.6. We derive next formulas for the second order derivatives involving a horizontal and a vertical directions. Lemma 3.3. With the assumptions of Theorem 1.3 we have (3.11 df(ξ i, I i Z = df(z 2n 3 T 0 (Z, f T 0 (I i Z, I i f 4( and (3.12 df(z, ξ i = df(i i Z n 1 T 0 (I i Z, f T 0 (Z, I i f 2n2 3n 1 U(Z, f ((n 1 ((n 1 U(I iz, f.

11 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 11 Proof. The second equality of (3.3 can be written in the form (3.13 ζ i (I i Z, f = df(z ( 2T (ξ i, I i Z, f ( 2 df(ξ i, I i Z 1 = df(z ( 2 T 4( 0 (Z, f T 0 (I i Z, I i f T (ξ s, I s Z, f df(ξ s, I s Z U(Z, f T 0 (Z, f 3U(Z, f (1 Sdf(Z n 1 n 1 U(Z, f ( 2 df(ξ i, I i Z, where we used (2.9 and (3.9. Now, equalities (3.13, (3.5 and the third equality in (2.10 imply (3.14 df(ξ i, I i Z = S 2n 3 df(z 2 4( = df(z 2n 3 4( Finally, the Ricci identity, (2.9 and (3.11 yield ( f(z, ξ i = df(ξ i, Z T (ξ i, Z, f = S 2 df(i iz which completes the proof. T 0 (Z, f T 0 (I i Z, I i f 1 U(Z, f ((n 1 T 0 (Z, f T 0 (I i Z, I i f 2n2 3n 1 U(Z, f. ((n 1 1 2( T 0 (I i Z, f n 1 T 0 (Z, I i f 2n2 n 2 = df(i i Z n 1 T 0 (I i Z, f T 0 (Z, I i f ((n 1 U(I iz, f ((n 1 U(I iz, f, Next, we compute the second vertical derivatives of f. We start with a basic useful identity involving only vertical derivatives. Lemma 3.4. With the assumptions of Theorem 1.3 the following identity holds ( f(ξ i, ξ i = f n 1 ( ea T 0 (e a, f ( ea T 0 (I i e a, I i f ( 1 ((n 1 ( e a U(e a, f. Proof. Differentiating (3.12, using (1.6 and (2.3 we obtain ( f(x, Y, ξ i α j (X 2 f(y, ξ k α k (X 2 f(y, ξ j = n 1 ( X T 0 (I i Y, f ( X T 0 (Y, I i f { f ω i (X, Y n 1 T 0 (X, I i Y T 0 (I i X, Y { df(ξ i g(x, Y n 1 { df(ξ j ω k (X, Y n 1 { df(ξ k ω j (X, Y n 1 α j (X df(i k Y n 1 T 0 (I i X, I i Y T 0 (X, Y T 0 (I j X, I i Y T 0 (I k X, Y T 0 (I k X, I i Y T 0 (I j X, Y T 0 (I k Y, f n 1 T 0 (Y, I k f α k (X df(i j Y n 1 T 0 (I j Y, f n 1 T 0 (Y, I j f ((n 1 ( XU(I i Z, f } ((n 1 U(X, I iy } ((n 1 U(X, Y } ((n 1 U(X, I ky ((n 1 U(X, I jy } ((n 1 U(I ky f ((n 1 U(I jy, f.

12 12 S. IVANOV, A. PETKOV, AND D. VASSILEV Applying again (3.12 to the second and the third terms in the first line we see that the terms involving the connection 1-forms cancel and (3.17 takes the following form ( f(x, Y, ξ i = n 1 f ( X T 0 (I i Y, f ( X T 0 (Y, I i f { ω i (X, Y n 1 { df(ξ i g(x, Y n 1 { df(ξ j ω k (X, Y n 1 { df(ξ k ω j (X, Y n 1 T 0 (X, I i Y T 0 (I i X, Y T 0 (I i X, I i Y T 0 (X, Y T 0 (I j X, I i Y T 0 (I k X, Y T 0 (I k X, I i Y T 0 (I j X, Y ((n 1 ( XU(I i Z, f } ((n 1 U(X, I iy } ((n 1 U(X, Y } ((n 1 U(X, I ky ((n 1 U(X, I jy On the other hand, the skew-symmetric part of (3.18 and the Ricci identity listed in the fourth line of (2.13 yield ( f(x, Y, ξ i 3 f(y, X, ξ i = n 1 ( X T 0 (I i Y, f ( X T 0 (Y, I i f ( Y T 0 (I i X, f ( Y T 0 (X, I i f ( X U(I i Y, f ( Y U(I i X, f 2f ω i (X, Y ((n 1 ((n 1 U(X, I iy { 2df(ξ j ω k (X, Y n 1 T 0 (I k X, Y T 0 } (X, I k Y ((n 1 U(X, I ky { 2df(ξ k ω j (X, Y n 1 T 0 (X, I j Y T 0 (I j X, Y = 2df(ξ j ρ k (X, Y 2df(ξ k ρ j (X, Y 2 ((n 1 U(X, I jy } }. ω s (X, Y 2 f(ξ s, ξ i. The trace X = e a, Y = I i e a of (3.19 and the second equality of (2.10 give (3.16, which completes the proof. Remark 3.5. The detailed proof of (3.18 shows a particular consequence of (2.3 which is that a covariant derivative of identities that are not Sp(1 invariant can lead to formulas which do not involve the connection one-forms. In the rest of the paper we shall usually skip many straightforward calculations some of which rely on a similar use of ( The elliptic eigenvalue problem. In this sub-section we will show that (1.6 implies that f satisfies an elliptic PDE. Let h be the Riemannian Laplacian of the metric (1.5. Lemma 3.6. On a qc manifold of dimension bigger than seven any smooth function satisfying (1.6 obeys the following identity (3.20 h f = ( 3f n 1 n( ( e a T 0 3 (e a, f ((n 1 ( e a U(e a, f. Proof. It is shown in 36, Lemma 5.1 that the Riemannian Laplacian h and the sub-laplacian of a smooth function f are connected by (3.21 h f = f 2 f(ξ s, ξ s. Equation (3.21 is a consequence of the formula (2.2, h f = a=1 h df(e a, e a 3 h df(ξ s, ξ s, and the identities T (e a, A, e a = T (ξ s, A, ξ s = 0, A Γ(T which follow from the properties of the torsion

13 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 13 tensor T of listed in (2.10. Lemma 3.4 and (2.8 imply ( f(ξ s, ξ s = 3f n 1 n( ( e a T 0 3 (e a, f ((n 1 ( e a U(e a, f. A substitution of (3.22 in (3.21, taking into account that f satisfies (1.6 hence f = f, we obtain (3.20 which proves the lemma. A consequence of Lemma 3.6 and Aronszajn s unique continuation result 1, is that f cannot vanish on any open set. We note this important fact in the next remark. Remark 3.7. If and f are as in Theorem 1.3 then f 0 in a dense set since f const Formulas for the torsion tensors. In this sub-section we derive formulas for the components T 0 and U of the torsion tensor. Lemma 3.8. With the assumption of Theorem 1.3 the following identities hold true for any X, Y, Z H (3.23 T 0 (I s f, I t f = 0, s t, s, t {1, 2, 3}, (3.24 T 0 ( f, f = 6n n 1 U( f, f, T 0 (I s f, I s f = 2n U( f, f, s {1, 2, 3}, n 1 (3.25 f 2 T 0 (Z, f = 6n U( f, fdf(z, n 1 f 2 U(Z, f = U( f, fdf(z, (3.26 f 4 T 0 (X, Y = 2n n 1 U( f, f 3df(Xdf(Y (3.27 f 4 U(Z, X = 1 ( n 1 U( f, f f 2 g(z, X n df(zdf(x df(i s Xdf(I s Y, df(i s Zdf(I s X. Proof. To determine the torsion tensors T 0 and U we are going to apply the following identity 39, 41 for the 1 component of the curvature (3.28 3R(Z, X, Y, f R(I 1 Z, I 1 X, Y, f R(I 2 Z, I 2 X, Y, f R(I 3 Z, I 3 X, Y, f = 2 g(x, Y T 0 (Z, f g(z, ft 0 (Y, X g(y, ZT 0 (X, f g( f, XT 0 (Y, Z 2 ω s (X, Y T 0 (Z, I s f ω s (Z, ft 0 (X, I s Y ω s (Z, Y T 0 (X, I s f ω s (X, ft 0 (Z, I s Y ( 2ω s (Z, X T 0 (Y, I s f T 0 (I s Y, f 8ω s (Y, fu(i s Z, X 4Sω s (Z, Xω s (Y, f. With the help of the Ricci identity, cf. the second equality of (2.13, we write the curvature tensor given by (3.2 in the form (3.29 R(Z, X, Y, f = df(zg(x, Y df(xg(z, Y df(z, ξ s ω s (X, Y df(x, ξ s ω s (Z, Y 2 df(ξ s, Y ω s (Z, X.

14 14 S. IVANOV, A. PETKOV, AND D. VASSILEV A calculation shows (3.30 R(I t Z, I t X, Y, f = t=1 g(x, Y s,t=1 df(i s Zω s (X, Y df(i s Xω s (Z, Y df(i t Z, ξ s ω s (I t X, Y df(i t X, ξ s ω s (I t Z, Y 2 df(ξ s, Y ω s (I t Z, I t X = df(i s Zω s (X, Y df(i s Xω s (Z, Y 2 df(ξ s, Y ω s (Z, X df(i s Z, ξ s g(z, Y ω i (X, Y df(i j Z, ξ k df(i k Z, ξ j df(i s X, ξ s (ijk ω i (Z, Y df(i j X, ξ k df(i k X, ξ j, (ijk where (ijk denotes the cyclic sum. Now, (3.29 and (3.30 together with (3.11 and (3.12 yield (3.31 3R(Z, X, Y, f R(I 1 Z, I 1 X, Y, f R(I 2 Z, I 2 X, Y, f R(I 3 Z, I 3 X, Y, f = g(x, Y 3df(Z 2 f(i s Z, ξ s g(z, Y 3df(X 2 f(i s X, ξ s 8 ω s (Z, X df(ξ s, Y { ω s (Z, X ω i (X, Y 3 2 f(z, ξ i df(i i Z 2 f(i j Z, ξ k 2 f(i k Z, ξ j (ijk ω i (Z, Y 3 2 f(x, ξ i df(i i X 2 f(i j X, ξ k 2 f(i k X, ξ j (ijk 4 = g(x, Y T 0 (Z, f 4 g(z, Y T 0 (X, f 4Sdf(I s Y 6 12n ((n 1 U(Z, f 12n ((n 1 U(X, f T 0 (I s Y, f T 0 (Y, I s f 8 } ((n 1 U(I sy, f { 4 ω s (X, Y T 0 } (Z, I s f ((n 1 U(I sz, f { 4 ω s (Z, Y T 0 } (X, I s f ((n 1 U(I sx, f. Subtracting (3.28 from (3.31 and applying (3.11, (3.12 and the properties of the torsion we come to ( = g(x, Y T 0 (Z, f 6n n 1 U(Z, f g(z, Y T 0 (X, f 6n n 1 U(X, f ω s (X, Y T 0 (Z, I s f ( 2n n 1 U(I sz, f ω s (Z, Y T 0 (X, I s f ω s (Z, X 2T 0 (I s Y, f 2T 0 (Y, I s f 4 n 1 U(I sy, f df(i s XT 0 (Z, I s Y df(i s ZT 0 (X, I s Y 4df(I s Y U(I s Z, X 2n n 1 U(I sx, f (df(xt 0 (Z, Y (df(zt 0 (X, Y.

15 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 15 Setting Z = f into (3.32, after some calculations, we obtain (3.33 ( f 2 T 0 (X, Y = (df(xt 0 ( f, Y g(x, Y T 0 ( f, f 6n n 1 U( f, f df(y T 0 (X, f 6n n 1 U(X, f df(i s Y T 0 (X, I s f 8n2 2n 4 U(I s X, f n 1 df(i s X 2T 0 (Y, I s f (2n 1T 0 (I s Y, f 4 n 1 U(I sy, f. Letting Y = f in (3.33, then using (3.34 and (3.6 shows (3.34 f 2 T 0 (X, f = T 0 3n ( f, fdf(x U( f, fdf(x f 2 U(X, f. (n 1(n 1 On the other hand, letting X = I 1 f in (3.33, using (3.6 and (3.34 gives ( = df(i 1 Y T 0 ( f, f T 0 (I 1 f, I 1 f 8n2 8n 4 U( f, f n 1 df(i 2 Y T 0 (I 1 f, I 2 f df(i 3 Y T 0 (I 1 f, I 3 f (2n 1 f 2 T 0 (Y, I 1 f T 0 (I 1 Y, f From (3.35 with Y = I 2 f and (3.8 the identity (3.23 follows since f 2 0. Setting Y = I 1 f into (3.35 implies (3.36 T 0 ( f, f T 0 (I 1 f, I 1 f = U( f, f. n 1 The latter equality together with the (2.8 yield (3.24. The equalities (3.35, (3.23 and (3.24 imply 4 n 1 f 2 U(I 1 Y, f. (3.37 (2n 1 f 2 T 0 (Y, I s f = (2n 1 f 2 T 0 (I s Y, f 4 n 1 f 2 U(I s Y, f df(i s Y T 0 ( f, f T 0 (I s f, I s f 8n2 8n 4 U( f, f n 1 = (2n 1 f 2 T 0 (I s Y, f 4 n 1 f 2 U(I s Y, f 4(df(I s Y U( f, f. Let Y = I 1 f in (3.32 in order to see (3.38 2( f 2 U(I 1 Z, X = ω 1 (Z, X T 0 ( f, f T 0 (I 1 f, I 1 f 2 n 1 U( f, f ndf(x T 0 (Z, I 1 f 1 n 1 U(I 1Z, f ndf(z T 0 (X, I 1 f 1 n 1 U(I 1X, f ndf(i 1 X T 0 (Z, f 3 n 1 U(Z, f ndf(i 1 Z T 0 (X, f 3 n 1 U(X, f ndf(i 2 X T 0 (Z, I 3 f 1 n 1 U(I 3Z, f ndf(i 2 Z T 0 (X, I 3 f 1 n 1 U(I 3X, f ndf(i 3 X T 0 (Z, I 2 f 1 n 1 U(I 2Z, f Letting X = f in (3.38 and applying (3.24 we obtain ndf(i 3 Z T 0 (X, I 2 f 1 n 1 U(I 2X, f (3.39 ( 2 n 2U(Z, f f 2 = (6n 2 n 2U( f, fdf(z n(n 1T 0 (I 1 Z, I 1 f f 2. Therefore, (3.40 f 2 n(n 1T 0 (Z, f 3( 2 n 2U(Z, f = 3(6n 2 n 2U( f, fdf(z..

16 16 S. IVANOV, A. PETKOV, AND D. VASSILEV On the other hand, taking into account (3.24, equality (3.34 yields (3.41 f 2 (n 2 1T 0 (Z, f 3nU(Z, f = 3n(U( f, fdf(z. Solving the system of equations (3.40 and (3.41, we obtain (3.25. A substitution of (3.25 and (3.43 in (3.33 gives (3.26. Now, a substitution of (3.25 and (3.43 in (3.38 shows (3.27. We finish this section with a few useful facts. As a direct corollary from (3.5, (3.25 and (3.43 it follows (3.42 f 2 (S 2 = In addition, from (3.25 and (3.37 it follows 4(n 1 U( f, f. n 1 (3.43 f 2 T 0 (Z, I s f = 2n n 1 U( f, fdf(i sz, f 2 T 0 (I s Y, I s f = 2n U( f, fdf(z. n 1 The equalities (3.25 and (3.43 yield (3.44 T 0 (I s Z, f = 3T 0 (Z, I s f, T 0 (Z, f = 3T 0 (I s Z, I s f, T 0 (Z, f = 6n U(Z, f, n Formulas for the covariant derivatives of the torsion tensors. Here we shall prove formulas for the covariant derivative of the torsion tensor. Lemma 3.9. If and f are as in Theorem 1.3, then we have the following identities for the covariant derivatives of the torsion tensor at the points where f 0, (3.45 f 2 ( Z T 0 (X, Y = 2 n 2 fdf(zt 0 (X, Y 2n n 1 U( f, f f 2 f 3df(Y g(x, Z 3df(Xg(Y, Z ( df(i s Y ω s (X, Z df(i s Xω s (Y, Z 2n U( f, f n 1 f 2 df(ξ i 3df(Y ω i (X, Z df(i i Y g(x, Z df(i j Y ω k (X, Z df(i k Y ω j (X, Z (ijk 2n U( f, f n 1 f 2 df(ξ i 3df(Xω i (Y, Z df(i i Xg(Y, Z df(i j Xω k (Y, Z df(i k Xω j (Y, Z and (ijk (3.46 f 2 ( Z U(X, Y 2fdf(ZU(X, Y 2 2 n n 1 df(ξ s df(i s ZU(X, Y 1 U( f, f n 1 f 2 U( f, f f 2 f df(y g(x, Z df(xg(y, Z df(ξ s df(i s ZU(X, Y = 2n 2 n 2 fdf(zu(x, Y 2fdf(Z 2 df(ξ s df(i s Z g(x, Y ( df(i s Y ω s (X, Z df(i s Xω s (Y, Z n U( f, f n 1 f 2 df(ξ i df(y ω i (X, Z df(i i Y g(x, Z df(i j Y ω k (X, Z df(i k Y ω j (X, Z (ijk n U( f, f n 1 f 2 df(ξ i df(xω i (Y, Z df(i i Xg(Y, Z df(i j Xω k (Y, Z df(i k Xω j (Y, Z, (ijk where (ijk means the cyclic sum, cf. Convention 1.4.

17 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 17 Proof. The contracted Bianchi identity reads 32, 41 (3.47 ( ea T 0 (e a, X 2n 4 n 1 ( e a U(e a, X (ds(x = 0. After taking the trace in the covariant derivatives of (3.25 and (3.42 we obtain (3.48 ( ea T 0 (e a, f = 6n ( U( f, f n 1 f f 2 22 n 1 ( ea U(e a, f = f The system (3.48 and (3.47 imply ( U( f, f U( f, f f 2 f f 2, f(s = 4 ( U( f, f n 1 f f 2. ( U( f, f (3.49 f f 2 Similarly, using in addition (3.44, we have (3.50 = 2 n 1 U( f, f f n 2 f 2. ( ea T 0 (e a, I s f = 2n ( U( f, f n 1 I s f f 2 ( ea U(e a, I s f = I s f I s f(s = 4 n 1 I s f f U( f, f f 2, 8n2 n 1 df(ξ U( f, f s f 2, ( U( f, f U( f, f f 2 df(ξ s f 2, ( U( f, f f 2. Since the differentiation of (3.43 involves covariant derivatives of the almost complex structures the derivation of (3.50 requires some care we do it explicitly again, cf. Remark 3.5. We start with the proof of the first formula in (3.50. Differentiating the first equation in (3.43, taking into account (2.3, we have ( X T 0 (Z, I i f α j (XT 0 (Z, I k f α k (XT 0 (Z, I j f T 0 (Z, I i X ( f ( U( f, f = 2n X n 1 f 2 U( f, f df(i i Z f 2 g ( X ( f, I i Z 2n U( f, f n 1 f 2 α j (Xdf (I k Z α k (Xdf (I j Z. The formula for the Hessian (1.6 gives ( X T 0 (Z, I i f α j (XT 0 (Z, I k f α k (XT 0 (Z, I j f ft 0 (I i X, Z = 2n X n 1 ( U( f, f f 2 ( U( f, f df(i i Z f 2 fg (X, I i Z 2n U( f, f n 1 f 2 α j (Xdf (I k Z α k (Xdf (I j Z. df (ξ s T 0 (I i I s X, Z df (ξ s g (I s X, I i Z Taking the trace in the above identity and then applying the first equation in (3.43 to the obtained equality we see that the terms involving the connection 1-forms cancel, which gives the first identity in (3.50. The second line in (3.50 follows similarly. The system (3.50 and (3.47 yields (3.51 I s f ( U( f, f f 2 = 2df(ξ s U( f, f f 2.

18 18 S. IVANOV, A. PETKOV, AND D. VASSILEV We calculate the divergence of T 0 differentiating (3.26, taking the trace in the obtained equality and applying (3.49, (3.51. After a short computation we obtain (3.52 f 2 ( ea T 0 (e a, Y 2fT 0 (Y, f 2 2n 3 f n 1 ( U( f, f 2n U( f, f n 1 f 2 f 2 df(ξ s T 0 (Y, I s f = df(y ( U( f, f I s f f 2 df(i s Y 2 f(e a, I s e a df(i s Y 3 2 f(e a, e a df(y 2n U( f, f n 1 f f( f, Y 2 f(i s f, I s Y. Applying (1.6, (3.25, (3.43, (3.49 and (3.51 to (3.52, we get (3.53 f 2 ( ea T 0 (e a, Y = 12n U( f, f f n 1 f 2 df(y U( f, f n 1 f 2 U( f, f n 1 f 2 df(ξ s df(i s Y 22 n 1 df(ξ s df(i s Y 12n U( f, f n 2 f 2 fdf(y U( f, f f 2 fdf(y 8n2 U( f, f n 1 f 2 6n U( f, f n 1 f 2 fdf(y 6n U( f, f n 1 f 2 2n U( f, f n 1 f 2 = df(ξ s df(i s Y df(ξ s df(i s Y fdf(y df(ξ i df(i i Y df(ξ j df(i j Y df(ξ k df(i k Y i=1 12n (n 1 ( U( f, f (n 2 (n 1 f 2 fdf(y n 1 U( f, f f 2 df(ξ s df(i s Y. Applying (3.25 and (3.43 to (3.53, we derive (3.54 f 2 ( ea T 0 (e a, Y = ( 2(n 1 n 2 ft 0 (Y, f ( 2 df(ξ s T 0 (Y, I s f.

19 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD 19 Now, we calculate the divergence of U differentiating (3.27, taking the trace in the obtained equality and applying (1.6, (3.49 and (3.51. We have (3.55 f 2 ( ea U(e a, Y 2fU(Y, f 2 1 ( U( f, f n 1 Y f 2 f 2 n f n 1 df(ξ s U(Y, I s f = ( U( f, f f 2 df(y 1 U( f, f n 1 f f(y, f n 2 f(e a, e a df(y n n U( f, f n 1 f 2 2 f( f, Y = 1 ( U( f, f n 1 Y f 2 f 2 2n n 1 n 1 2n fu(y, f 2n n 1 n 1 2 f(i s f, I s Y n 2 fu(y, f ( U( f, f I s f f 2 df(i s Y 2 f(e a, I s e a df(i s Y df(ξ s U(I s Y, f 2 2 fu(y, f n 1 df(ξ s U(I s Y, f df(ξ s U(I s Y, f. Thus, from (3.55 we obtain (3.56 f 2 ( ea U(e a, Y = 1 ( U( f, f n 1 Y f 2 f 2 2 6n n 2 fu(y, f 2 2n n 1 df(ξ s U(I s Y, f. A substitution of (3.54, (3.56 and f 2 Y (S = 4 ( U( f, f n 1 Y f 2 f 2 in (3.47 implies ( U( f, f (3.57 Y f 2 f 2 = 2n 2 n 2 fu(y, f 2 df(ξ s U(I s Y, f. The equalities (3.56 and (3.57 yield (3.58 f 2 ( ea U(e a, Y = 2(n 1( n 2 fu(y, f 2( df(ξ s U(I s Y, f.

20 20 S. IVANOV, A. PETKOV, AND D. VASSILEV We calculate from (3.26 using (1.6, (3.25 and (3.57 that f 2 ( Z T 0 (X, Y 2fdf(ZT 0 (X, Y 2 df(ξ s df(i s ZT 0 (X, Y = 2n 2 n 2 fdf(zt 0 (X, Y 2 2n n 1 U( f, f f 2 f df(ξ s df(i s ZT 0 (X, Y 3df(Y g(x, Z df(i s Y ω s (X, Z 2n U( f, f n 1 f 2 f 3df(Xg(Y, Z df(i s Xω s (Y, Z 2n U( f, f n 1 f 2 df(ξ 1 3df(Y ω 1 (X, Z df(i 1 Y g(x, Z df(i 2 Y ω 3 (X, Z df(i 3 Y ω 2 (X, Z 2n U( f, f n 1 f 2 df(ξ 1 3df(Xω 1 (Y, Z df(i 1 Xg(Y, Z df(i 2 Xω 3 (Y, Z df(i 3 Xω 2 (Y, Z 2n U( f, f n 1 f 2 df(ξ 2 3df(Y ω 2 (X, Z df(i 2 Y g(x, Z df(i 1 Y ω 3 (X, Z df(i 3 Y ω 1 (X, Z 2n U( f, f n 1 f 2 df(ξ 2 3df(Xω 2 (Y, Z df(i 2 Xg(Y, Z df(i 1 Xω 3 (Y, Z df(i 3 Xω 1 (Y, Z 2n U( f, f n 1 f 2 df(ξ 3 3df(Y ω 3 (X, Z df(i 3 Y g(x, Z df(i 1 Y ω 2 (X, Z df(i 2 Y ω 1 (X, Z 2n U( f, f n 1 f 2 df(ξ 3 3df(Xω 3 (Y, Z df(i 3 Xg(Y, Z df(i 1 Xω 2 (Y, Z df(i 2 Xω 1 (Y, Z. The last equality yields (3.45. Finally, equation (3.46 follows from (3.27 using (1.6, (3.25 and (3.57. In the next, key step of the proof, where we show that the torsion tensor vanishes, we shall use the following particular cases of Lemma 3.9. For Z = f, (3.45 gives (3.59 ( f T 0 (X, Y = 2n 2 n 2 ft 0 (X, Y df(ξ 1 T 0 (I 1 X, Y T 0 (X, I 1 Y df(ξ 2 T 0 (I 2 X, Y T 0 (X, I 2 Y df(ξ 3 T 0 (I 3 X, Y T 0 (X, I 3 Y. Similarly, letting Z = I i f in (3.45 we obtain (3.60 ( Ii f T 0 (X, Y = 2df(ξ i T 0 (X, Y f T 0 (I i X, Y T 0 (X, I i Y df(ξ j T 0 (I k X, Y T 0 (X, I k Y df(ξ k T 0 (I j X, Y T 0 (X, I j Y. The substitution of Y = f in (3.54 taking into account (3.24 and Lemma 3.2 gives (3.61 f 2 ( ea T 0 (e a, f = 12n(n 1( fu( f, f, (n 2(n 1 while the substitution Z = e a, X = I i e a, Y = I i f in (3.45 and (3.43 gives (3.62 f 2 ( ea T 0 (I i e a, I i f = Finally, letting Z = f, I s f in (3.46 shows the next equalities (n 1 ( fu( f, f. (n 2 (n 1 (3.63 ( f U(X, Y = 2n 2 n 2 fu(x, Y, ( I s f U(X, Y = 2df(ξ s U(X, Y.

21 THE OBATA SPHERE THEORES ON A QUATERNIONIC CONTACT ANIFOLD Vanishing of the torsion. In this section we show the vanishing of the torsion, T 0 = U = 0, by calculating in two ways the mixed third covariant derivatives of a function satisfying (1.6. Lemma If satisfies the assumptions of Theorem 1.3, then the torsion tensor vanishes, T 0 = 0, U = 0, i.e., is a qc-einstein manifold. The proof occupies the remaining part of this sub-section The Ricci identities. We are going to use the sixth line in (2.13. A substitution of the contracted Bianchi identity (3.47 in the second formula of (2.12 gives (3.64 (ρ i (ξ j, I k X = (ρ i (ξ k, I j X = 1 4 ( e a T 0 (e a, X ( ea T 0 (I i e a, I i X n n 1 ( e a U(e a, X. Let Z = f in (2.11, and then substitute the obtained equality in the sixth formula of (2.13, after which use (3.64 in order to see ( f(ξ i, X, Y 3 f(x, Y, ξ i = 2 f(t (ξ i, X, Y 2 f(x, T (ξ i, Y df(( X T (ξ i, Y ( X U(I i Y, f 1 ( Y T 0 (I i f, X ( Y T 0 ( f, I i X 1 ( f T 0 (I i Y, X ( f T 0 (Y, I i X ( e a T 0 (e a, I k f (I k e a, f n n 1 ( e a U(e a, I k f ω j (X, Y ( e a T 0 (e a, I j f (I j e a, f n n 1 ( e a U(e a, I j f ω k (X, Y ( e a T 0 (e a, I k Y (I k e a, Y n n 1 ( e a U(e a, I k Y df(i j X ( e a T 0 (e a, I j Y (I j e a, Y n n 1 ( e a U(e a, I j Y df(i k X ( e a T 0 (e a, I k X (I k e a, X n n 1 ( e a U(e a, I k X df(i j Y ( e a T 0 (e a, I j X (I j e a, X n n 1 ( e a U(e a, I j X df(i k Y. Note that from (2.9 we have (3.66 T (ξ i, X, Y = 1 4 T 0 (I i X, Y T 0 (X, I i Y U(X, I i Y. Differentiating the above formula we find, applying (2.3, (3.67 df (( X T (ξ i, Y = 1 4 ( X T 0 (I i Y, f ( X T 0 (Y, I i f ( X U(I i Y, f.

22 22 S. IVANOV, A. PETKOV, AND D. VASSILEV Using (1.6, (3.66, (3.67 and the properties of torsion tensor listed in (2.8, we obtain from (3.65 ( f(ξ i, X, Y 3 f(x, Y, ξ i = 1 ( X T 0 (I i Y, f ( X T 0 (Y, I i f 1 ( Y T 0 (I i X, f ( Y T 0 (X, I i f ( f T 0 (X, I i Y ( f T 0 (I i X, Y 1 T 0 (I i X, Y T 0 (X, I i Y f T 0 (X, I k Y T 0 (I k X, Y 2U(X, I k Y df(ξ j T 0 (I j X, Y T 0 (X, I j Y 2U(I j X, Y df(ξ k ( e a T 0 (e a, I k f (I k e a, f n n 1 ( e a U(e a, I k f ω j (X, Y ( e a T 0 (e a, I j f (I j e a, f n n 1 ( e a U(e a, I j f ω k (X, Y ( e a T 0 (e a, I k Y (I k e a, Y n n 1 ( e a U(e a, I k Y df(i j X ( e a T 0 (e a, I j Y (I j e a, Y n n 1 ( e a U(e a, I j Y df(i k X ( e a T 0 (e a, I k X (I k e a, X n n 1 ( e a U(e a, I k X df(i j Y ( e a T 0 (e a, I j X (I j e a, X n n 1 ( e a U(e a, I j X df(i k Y. For X = I i f, Y = f equation (3.68 together with (3.59, (3.60, (3.6 and (3.23 imply ( f(ξ i, I i f, f 3 f(i i f, f, ξ i = 0. On the other hand, a subtraction of (3.18 from (3.1 with A = ξ i gives ( f(ξ i, X, Y 3 f(x, Y, ξ i = n 1 ( X T 0 (I i Y, f ( X T 0 (Y, I i f ((n 1 ( XU(I i Y, f f n 1 T 0 (X, I i Y T 0 (I i X, Y ((n 1 fu(x, I iy df(ξ i n 1 T 0 (I i X, I i Y T 0 (X, Y ((n 1 df(ξ iu(x, Y df(ξ j n 1 T 0 (I j X, I i Y T 0 (I k X, Y ((n 1 df(ξ ju(x, I k Y df(ξ k n 1 T 0 (I k X, I i Y T 0 (I j X, Y ((n 1 df(ξ ku(x, I j Y 2 f(ξ i, ξ s f ω s (X, Y. Letting X = I i f, Y = f in (3.70 and then applying (3.6 and (3.23 we obtain ( f(ξ i, I i f, f 3 2(n 1 f(i i f, f, ξ i = ( I i f T 0 (I i f, f ((n 1 ( I i f U(I i f, f n 1 f T 0 (I i f, I i f T 0 ( f, f ((n 1 fu( f, f 2 f(ξ i, ξ i f f 2.