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.
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 co…
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…
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 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 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.
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.
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 …
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.
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 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.
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 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 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 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 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.
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.
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…
The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kernels are also discussed and the connection to the quaternionic magnetic Laplacian is done.
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
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…
The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.
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.
On the seven dimensional Euclidean sphere $\mathbb{S}^7$ we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution $\mathcal{H}_T$ is induced by a Clifford module structure of $\mathbb{R}^8$. More precisely, $\mathcal{H}_T$ is rank $4$, bracket generating of step two and generated by globally defined vector fields. The distribution $\mathcal{H}_{Q}$ of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fi…
In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants $χ$ and $κ$ defined on a 3D contact structure.
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…
We extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds [arXiv:0806.4620] to the case of quaternionic-Kahler manifolds. Via Swann's construction, deformations of a 4d-dimensional quaternionic-Kahler manifold $M$ are in one-to-one correspondence with deformations of its $4d+4$-dimensional hyperkahler cone $S$. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space $Z_S$, with a suitable homogeneity condition that ensures that the hyperkahler cone property is pr…
We show that $3$-Sasaki structures admit a natural description in terms of projective differential geometry. This description provides a concrete link between $3$-Sasaki structures and several other geometries and constructions via a single unifying picture. First we establish that a $3$-Sasaki structure may be understood as a projective structure equipped with a certain holonomy reduction to the (possibly indefinite) unitary quaternionic group $\textrm{Sp}(p,q)$, namely a parallel hyperkähler structure on the projective tractor bundle satisfying a particular genericity condition. For the conv…
In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the newly defined classes should admit 'good' metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-$(α,δ)$-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heis…
We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. In particular we prove that contact versions of the relative eta-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.
In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are $A_3$ and $A_5$ (in the classification of Arnol'd's school). We show that in the non-gene…
We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian; 4) Spectral gap estimates.
We find that the target space of two-dimensional (4,0) supersymmetric sigma models with torsion coupled to (4,0) supergravity is a QKT manifold, that is, a quaternionic Kähler manifold with torsion. We give four examples of geodesically complete QKT manifolds one of which is a generalisation of the LeBrun geometry. We then construct the twistor space associated with a QKT manifold and show that under certain conditions it is a Kähler manifold with a complex contact structure. We also show that, for every 4k-dimensional QKT manifold, there is an associated 4(k+1)-dimensional hyper-Kähler one.
