arXiv research

The article proves a Hineva inequality for various submanifolds in Quaternionic Space forms.

Paper studies quaternionic space forms and Riemannian maps inequalities.

In this paper we establish some inequalities concerning the $k$-Ricci curvature of a slant submanifold in a quaternionic space form. We also obtain obstructions to the existence of quaternionic slant immersions in quaternionic space forms with unfull first normal bundle.

The paper proves quaternion projective space is unstable.

The paper derives curvature inequalities for submersions from quaternionic space forms.

This is a complete classification of the complex forms of quaternionic symmetric spaces

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…

In this paper, we introduce the quaternionic similar curves in 4-dimensional Euclidean space. We show that the families of quaternionic curves with vanishing curvatures form the families of quaternionic similar curves.

It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. …

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

We prove that a compact quaternionic-Kähler manifold of dimension $4n\geq 8$ admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionic-Kähler structure.

In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized $δ$-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent …

The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.

We prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which, to any pair formed of a quaternionic vector space and a real subspace, associates a coherent sheaf over the sphere.

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connect…

We investigate the integrability of almost complex structures on the twistor space of an almost quaternionic manifold constructed with the help of a quaternionic connection. We show that if there is an integrable structure it is independent on the quaternionic connection. In dimension four, we express the anti-self-dua…

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 study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle o…

We study the relations between the quaternion $H$-type group and the boundary of the unit ball on two dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion $H$-type group into its subspace of boundary values of $q$-holomorphic functions is consider. …

Study geodesic orbit metrics in quaternionic Stiefel manifolds.

A 2-form on a quaternionic-Kahler manifold (M, g) is called compatible (with the quaternionic structure) if it is a section of the direct sum bundle S^2(H) \oplus S^2(E). We construct a connection D on S^2(H) \oplus S^2(E)\oplus TM, which is a prolongation of the conformal-Killing operator acting on compatible 2-forms.…

This paper studies geometric structures on manifolds with specific symplectic properties.

Using quaternionic Feix--Kaledin construction we provide a local classification of quaternion-Kähler metrics with a rotating $S^1$-symmetry with the fixed point set submanifold $S$ of maximal possible dimension. For any Kähler manifold $S$ equipped with a line bundle with a unitary connection of curvature proportional …

It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider the case of tangent (sphere) bundle over the real, complex and quaternionic space …

We describe the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kaehler metric on the simply-connected 8-manifold G_2/SO(4) that carry a closed fundamental 4-form but are not Einstein.

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvat…

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge^2 W \to V. If the skew symmetric vector valued bilinear form Πis nondegenerate then (M,Q) i…

The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.

Researchers find a way to estimate potential functions for quaternionic metrics.

We prove that if the fundamental 4-form of an almost-quaternionic Hermitian manifold (M, Q, g) of dimension at least eight satisfies the conformal-Killing equation, then (M, Q, g) is quaternionic-Kahler.

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…

The purpose of this note is to define tri-moment maps for certain manifolds that carry closed non-degenerate 4-forms and an $Sp(1)^n$-action. Examples include quaternionic vector spaces and flag manifolds. We show how this map can be used ro reduce such manifolds to the ones with fewer symmetries. The images of such ma…

We prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.

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

The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.

We show that every Killing p-form on a compact quaternion-Kähler manifold has to be parallel for p greater than 1.

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.

The study finds conditions for quaternionic structures on symmetric spaces.

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines…

The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.

We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in…

We consider a general 4n-dimensional quaternionic Kahler geometry with a free action of the torus T^(n+1). The toric action lifts onto the Swann bundle of the quaternionic Kahler space to a tri-holomorphic action that commutes with the standard H* action on the bundle. By matching Pedersen and Poon's generalized Gibbon…

Quaternionic differential geometry expands geometric concepts using quaternions.

The paper extends Gray's result to quaternion-Kähler manifolds.

DeTurck and Yang have shown that in the neighbourhood of every point of a $3$-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, whith respect to which the metric has diagonal form). We show that this property does not generalize to higher dimensions. In particular, the complex p…

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…

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…