arXiv research

Study local control in a 7D quaternionic Heisenberg group.

In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heise…

The paper studies metrics and geodesics on a quaternionic Heisenberg group.

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quat…

We determine the best (optimal) constant in the $L^2$ Folland-Stein inequality on the quaternionic Heisenberg group and the non-negative functions for which equality holds.

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.

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 sp…

The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the E…

New method constructs degenerate Sasakian manifolds from hyperkähler bundles.

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 integra…

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.

Study of symmetries in deformed q-map spaces reveals a complex group structure.

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. W…

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

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 quaternion…

Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.

New smooth solutions of the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton based on the quaternionic Heisenberg group are constructed. We show that through appropriate contractions the solutions found in the $G_2$-heterotic case converge to the heterotic solutions on 6-dimensi…

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 con…

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 an…

The paper studies quaternionic Kähler manifolds and their fibration by solvmanifolds.

The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on $\mathbb{S}^3$ a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than tho…

The paper explores automorphism groups of parabolic structures on aspherical manifolds.

We show that if the lower central series of the fundamental group of a closed oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental group…

Local flatness theorem for paraquaternionic contact structures.

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient co…

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 …

Let $M^{4n}$ be a complete quaternionic Kähler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $λ_1(M)$ of the Laplacian which is $λ_1(M)\le (2n+1)^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\mathbb{R}\times N$ w…

Study magnetic fields on special Lie groups, proving non-existence of certain types.

New conformally Einstein metrics on Heisenberg group found.

Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.

Classifies geodetically convex sets and functions on Heisenberg group.

Homogeneous magnetic paths found in Heisenberg space.

Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.

Find conditions for starshapedness of level sets in Heisenberg group.

Defines contact structures on Heisenberg groups for geometric interpretation.

Computes minimal polynomials for generalized Heisenberg groups.

The paper describes geodesics on a Kähler cone of the Heisenberg group.

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

In Heisenberg group, bisectors are spinal spheres with specific curvature.

Criterion for surfaces in Heisenberg group to be graphs using flat cones.

Translation surfaces in Heisenberg group classified by Gauss map determinant.

Study maps surface configurations to Heisenberg homologies for mapping class groups.

Among eight possible geometric structures on three-dimensional manifolds less studied from the differential geometric point of view are those modelled on the Heisenberg group $Heis^3$. We consider the Heisenberg left-invariant metric and use some results on Levi-Civita connection and curvature tensor to present solutio…

Study shows only hyperplanes in Heisenberg groups have zero curvature.

Develops analysis of Hölder continuous mappings on Heisenberg groups.

In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmon…

Study on Killing magnetic curves in Heisenberg group geometry.

Sharp stability of isometries on Heisenberg group proven.