A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All conformal deformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing torsion of the Biquard connection are explicitly described. A '3-Hamiltonian form' of infinitesimal conformal automorphisms of quaternionic contact structures is presented.
A complete solution to the quaternionic contact Yamabe equation on the qc sphere of dimension $4n+3$ as well as on the quaternionic Heisenberg group is given. A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown.
A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the $L^2$ Folland-Stein embedding theorem is determined.
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
It is shown that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Namely, it is proved that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure o…
We show that the fundamental 4-form on a quaternionic contact manifold of dimension at least eleven is closed if and only if the torsion endomorphism of the Biquard connection vanishes. This condition characterizes quaternionic contact structures which are locally qc homothetic to 3-Sasakian structures.
A quaternionic contact (qc) heat equation and the corresponding qc energy functional are introduced. It is shown that the qc energy functional is monotone non-increasing along the qc heat equation on a compact qc manifold provided certain positivity conditions are satisfied.
We propose a global invariant $σ_c$ for contact manifolds which admit a strictly pseudoconvex CR structure, analogous to the Yamabe invariant $σ$. We prove that this invariant is non-decreasing under handle attaching and under connected sum. We then give a lower bound on $σ_c$ in a particular case.
We show that a compact quaternionic contact manifold of dimension seven that satisfies a Lichnerowicz-type lower Ricci-type bound and has the $P$-function of any eigenfunction of the sub-Laplacian non-negative achieves its smallest possible eigenvalue only if the structure is qc-Einstein. In particular, under the stated conditions, the lowest eigenvalue is achieved if and only if the manifold is qc-equivalent to the standard $3$-Sasakian sphere.
The main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the $Sp(n)Sp(1)$ components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a-priori estimates for square integrals of horizontal derivatives of …
In this paper, we study the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients $c_0$ and $c_1$ appearing in the small time asymptotics expansion of the heat kernel on the diagonal. We show that the second coefficient $c_1$ depends linearly on the qc scalar curvature $κ$. Finally we apply our results to compact qc-Einstein manifolds and prove the spectral invariance of geometric quantities in the subriemannian setting.
Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, its Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe in…
We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(α,β)$-Ricci-Yamabe soliton with $α\neq 0$ and potential vector field collinear with the Reeb vector field $ξ$ on a complete contact metric manifold with the Reeb vector field $ξ$ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $ξ$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $α\neq 0$, then …
We prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.
This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the torsion and curvature at the center point.
It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n + 1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting…
The conformal infinity of a quaternionic-Kahler metric on a 4n-manifold with boundary is a codimension 3-distribution on the boundary called quaternionic contact. In dimensions 4n-1 greater than 7, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kahler metric. On the contrary, in dimension 7, we prove a criterion for quaternionic contact structures to be the conformal infinity of a quaternionic- Kahler metric. This allows us to find the quaternionic-contact structures on the 7-sphere close to the conformal infinity of the quaternionic hyperbolic metric and w…
We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n + 1. The problem of finding solutions with minimum energy has been resolved for all dimensions except dimension 5 (n = 2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.
We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.
We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kaehler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kaehler metrics and Spin(7)-holonomy metrics which seem to be new. Moreover, we give expl…
We exploit the Cartan-Kähler theory to prove the local existence of real analytic quaternionic contact structures for any prescribed values of the respective curvature functions and their covariant derivatives at a given point on a manifold. We show that, in a certain sense, the different real analytic quaternionic contact geometries in $4n+3$ dimensions depend, modulo diffeomorphisms, on $2n+2$ real analytic functions of $2n+3$ variables.
In this note, we prove that the CR manifold which is induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.
Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components.
The main result is that the qc-scalar curvature of a seven dimensional quaternionic contact Einstein manifold is a constant. In addition, we characterize qc-Einstein structures with certain flat vertical connection and develop their local structure equations. Finally, regular qc-Ricci flat structures are shown to fibre over hyper-Kahler manifolds.
An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator, and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positiv…
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
In this paper we establish an analogue of the classical Lichnerowicz' theorem giving a sharp lower bound of the first non-zero eigenvalue of the sub-Laplacian on a compact seven-dimensional quaternionic contact manifold, assuming a lower bound of the qc-Ricci tensor, torsion tensor and its distinguished covariant derivatives.
We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show that an embedded qc-hypersurface in a hyper-Kähler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-Kähler metric is degenerate along the hypersurface.
We establish quaternionic contact (qc) versions of the so called Almost Schur Lemma, which give estimations of the qc scalar curvature on a compact qc manifold to be a constant in terms of the norm of the $[-1]$-component and the norm of the trace-free part of the $[3]$-component of the horizontal qc Ricci tensor and the torsion endomorphism, under certain positivity conditions.
A version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(1)Sp(1) component of the qc-Ricci curvature on a compact seven dimensional quaternionic contact manifold is established. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached if and only if the quaternionic contact manifold is a round 3-Sasakian sphere.
We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kähler manifold is endowed with an involutive 7-dimensional distribution whose integral leafs are locally qc-conformal to the standard 3-Sasakian sphere.
We show that the CR structure on the twistor space of a quaternionic contact structure described by Biquard is normal if and only if the Ricci curvature of the Biquard connection commutes with the endomorphisms in the quaternionic structure of the contact distribution.
We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. We present a left invariant quaternionic contact structure on a seven dimensional non-nilpotent Lie group, and show that this structure is locally quaternionic contact conformal to the flat quaternionic contact structure on the quaternionic Heisenberg group. On the product of a se…
We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal manifolds with symplectic conformal holonomy of the appropriate signature; as pseudo-Riemannian manifolds admitting conformal Killing fields satisfying a conformally invariant system of conditions analog to G. Sparling's criteria; and as the total space of a SO(3)- or $S^3$-bundle over a qc manifold with the conformally equivalent metrics defined directly by Biqu…
We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.
We prove a quaternionic contact versions of the Obata's sphere theorems. We show that if the first positive eigenvalue of the sub-Laplacian on a compact qc manifold of dimension bigger than seven 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. We also give a version of the theorem on non-compact qc manifold which is complete with respect to the associated Riemannian metric using the existence of a function with traceless horizontal Hessian. A qc version of the Liouville theorem is shown for qc-confo…
We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.
Let Z be a compact complex (2n+1)-manifold which carries a {\em complex contact structure}, meaning a codimension-1 holomorphic sub-bundle D of TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M^{4n}, g). If Z also admits a second complex contact structure, then Z= CP_{2n+1}. As an application, we give several new characterizations of the Riemannian manifold HP_n=Sp(n+1)/( Sp(n)\times Sp(1)).
We apply the theory of Weyl structures for parabolic geometries developed by A. Cap and J. Slovak in to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot-Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold. In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern-Moser tensor in CR geometry. This ten…
A non-convergence phenomenon for the CR Yamabe flow
Université de Montpellier, Institut Montpelliérain Alexander Grothendieck（蒙彼利埃大学，亚历山大·格罗滕迪克蒙彼利埃研究所）；Tamkang University（淡江大学）；Università di Trento（特伦托大学）
Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster connection in complex CR geometry, we construct a canonical connection associated with a quaternionic pseudohermitian structure, when the underlying quaternionic CR structure satisfies the ultra-pseudoconvexity which is stronger than the strong pseudoconvexity. Comparison to Biquard…
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvat…
In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost $3$-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the $3$-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact structure of the quaternionic Heisenberg groups. We focus on t…
We apply the general theory of codimension one integrability conditions for $G$-structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold.
Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of the family of solutions of a system of an odd number of second order ODE's subject to a single maximally non-integrable constraint. A subclass of contact path geometries is distinguished by the vanishing of an invariant contact torsion. For this subclass the equivalence problem i…
Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-negative and the Webster curvature is positive. Our lower bound for the non-zero eigenvalues is sharp and is attained on S^3. A consequence of our lower bound is that all compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable. N…
Unit tangent bundles $UM$ of semi-Riemannian manifolds $M$ are shown to be examples of dynamical Legendrian contact structures, which were defined in recent work [25] of Sykes-Zelenko to generalize leaf spaces of 2-nondegenerate CR manifolds. In doing so, Sykes-Zelenko extended the classification in Porter-Zelenko [20] of regular, 2-nondegenerate CR structures to those that can be recovered from their leaf space. The present paper treats dynamical Legendrian contact structures associated with 2-nondegenerate CR structures which were called "strongly regular" in Porter-Zelenko, named "L-contact…
In this work, we prove that every complex contact structure gives rise to a distinguished type of almost contact metric 3-structure. As an application of our main result, we provide several new examples of manifolds which admit taut contact circles, taut and round almost cosymplectic 2-spheres, and almost hypercontact (metric) structures. These examples generalize, in a suitable sense, the well-known examples of contact circles defined by the Liouville-Cartan forms on the unit cotangent bundle of Riemann surfaces. Furthermore, we provide sufficient conditions for a compact complex contact mani…
