We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold $N$, such that $TM$ generates a quaternionic subbundle of $TN|_M$, of (real) rank $4k$, there exists, locally, a quaternionic submanifold of $N$, containing $M$ as a hypersurface.

2011-09-30math.DG

A note on first eigenvalue estimates by coupling methods in Kähler and quaternion Kähler manifolds

In this short note, using the Kendall-Cranston coupling, we study on Kähler (resp. quaternion Kähler) manifolds first eigenvalue estimates in terms of dimension, diameter, and lower bounds on the holomorphic (resp. quaternionic) sectional curvature.

2021-12-08math.DGmath.PR

On the twistor space of a quaternionic contact manifold

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.

2011-04-14math.DG

Notes on Spinors and Polyforms II: Quaternions and Octonions

Pauli matrices are 2x2 tracefree matrices with a real diagonal and complex (complex-conjugate) off-diagonal entries. They generate the Clifford algebra Cl(3). They can be generalised by replacing the off-diagonal complex number by one taking values in either quaternions or octonions (or their split versions). These quaternionic and octonionic generalisations generate well-known models of Cl(5) and Cl(9) respectively. The main aim of the paper is to explicitly relate these models to the models arising via the creation/annihilation operator construction. We describe in details the models related…

2022-05-10hep-thmath-phmath.DG

关于四元数射影空间的稳定性

On the stability of the quaternion projective space

本文证明了负常四元数截面曲率下四元数空间形式恒等映射指标为零，得出紧致负曲率四元数空间形式稳定，而四元数射影空间不稳定的结论。

2025-08-12math.DG

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A note on the $\hat A$-genus for $π_2$-finite manifolds with $S^1$-symmetry

We construct examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. This is related to the classification of positive quaternionic Kaehler manifolds.

2008-11-05math.DG

Tetraplectic structures, tri-momentum maps, and quaternionic flag manifolds

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 maps for quaternionic flag manifolds, which are defined using the Dieudonné determinant, resemble the polytopes from the complex case.

2001-06-25math.DGmath.SG

A note on a Sung-Wang's paper

The purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $λ_{1,p}$ for the $p$-Laplacian achievies its maximal value on a Kähler manifold or a quaternionic Kähler manifold then such a manifold must be connected at infinity unless it is a topological cylinder with an explicit warped product metric.

2015-05-28math.DG

Para-quaternionic reduction

The pseudo-Riemannian manifold $M=(M^{4n},g), n \geq 2$ is para-quaternionic K\" ahler if $hol(M) \subset sp(n, \RR) \oplus sp(1, \RR).$ If $hol(M) \subset sp(n, \RR),$ than the manifold $M$ is called para-hyperK\" ahler. The other possible definitions of these manifolds use certain parallel para-quaternionic structures in $\End (TM),$ similarly to the quaternionic case. In order to relate these different definitions we study para-quaternionic algebras in details. We describe the reduction method for the para-quaternionic K\" ahler and para-hyperK\" ahler manifolds and give some examples. The …

2003-04-27math.DG

Bonnet pairs of surfaces in Minkowski space

In this note we consider the problem of local classification of Bonnet pairs of surfaces in 3-dimensional Minkowski space. We use split quaternions in a way similar to the use of quaternions in, arXiv:dg-ga/9610006, for the solution in Euclidean 3-Space.

2012-05-03math.DG

Non-degenerate homogeneous $ε$-Kähler and $ε$-quaternion Kähler structures of linear type

We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respectively. In addition the corresponding homogeneous models are computed, exhibiting the relation between these kind of structures and the incompleteness of the metric.

2013-11-13math.DG

About the geometry of almost para-quaternionic manifolds

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is…

2008-08-19math.DG

Homogeneous structures of linear type on $ε$-Kähler and $ε$-quaternion Kähler manifolds

We analyze degenerate homogeneous structures of linear type in the pseudo-Kähler and para-Kähler cases. The local form and the holonomy of pseudo-Kähler or para-Kähler manifolds admitting such structure are obtained. In addition the associated homogeneous models are studied exhibiting their relation with the incompleteness of the metric. The same questions are tackled in the pseudo-quaternion Kähler and para-quaternion Kähler cases. These results complete the study of homogeneous structures of linear type in pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases.

2013-10-16math.DG

Interactions between para-quaternionic and Grassmannian geometry

Almost para-quaternionic structures on smooth manifolds of dimension $2n$ are equivalent to almost Grassmannian structures of type $(2,n)$. We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view. In particular, we relate the respective normalization conditions, distinguished curves and twistor constructions.

2018-10-29math.DG

Metric geometries over the split quaternions

We give an overview of some recent results in hypersymplectic and para-quaternionic Kahler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kahler manifolds. These are used to classify examples with a fully homogeneous action of a semi-simple Lie group, and to construct distinct para-quaternionic Kahler metrics from indefinite real analytic conformal manifolds. We also indicate how the theory of toric varieties gives rise to constructions of hypersymplectic manifolds.

2004-12-10math.DG

A note on invariant description of $SU(2)$-structures in dimension 5

We develop an invariant approach to $SU(2)$--structures on spin $5$--manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to $SU(2)$. Moreover, we show how to induce quaternionic structure on the contact distribution of considered $SU(2)$--structure. We conclude with the invariance of certain components of the covariant derivative $\nablaφ$, where $φ$ is any spinor field defining considered $SU(2)$--structure. This shows, what expected, that (at least some of) the intrinsic torsion modules, can be derived invariantly with th…

2021-12-28math.DG

Killing forms on Quaternion-Kähler manifolds

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

2008-10-03math.DG

Ricci flow unstable cell centered at an Einstein metric on the twistor space of positive quaternion Kähler manifolds of dimension $\geq 8$

We show that a 1-parameter family Ricci flow ancient solutions arises from the natural collapsings of the twistor space of positive quaternion Kähler manifolds. We use these ancient solutions to show that a positive quaternion Kähler manifold is isometric to one of the Wolf spaces.

2008-10-14math.DG

(Para-)Hermitian and (para-)Kähler Submanifolds of a para-quaternionic Kähler manifold

On a para-quaternionic Kähler manifold $(\widetilde M^{4n},Q,\widetilde g)$, which is first of all a pseudo-Riemannian manifold, a natural definition of (almost) Kähler and (almost) para-Kähler submanifold $(M^{2m},\mathcal{J},g)$ can be given where $\mathcal{J}=J_1|_M$ is a (para-)complex structure on $M$ which is the restriction of a section $J_1$ of the para-quaternionic bundle $Q$. In this paper, we extend to such a submanifold $M$ most of the results proved by Alekseevsky and Marchiafava, 2001, where Hermitian and Kähler submanifolds of a quaternionic Kähler manifold have been studied. Co…

2012-01-19math.DG

Twistor and Reflector Spaces of Almost Para-Quaternionic Manifolds

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 connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.

2005-11-27hep-thmath-phmath.DG

The para-HK/QK correspondence

We generalise the hyper-Kahler/quaternionic Kahler (HK/QK) correspondence to include para-geometries, and present a new concise proof that the target manifold of the HK/QK correspondence is quaternionic Kahler. As an application, we construct one-parameter deformations of the temporal and Euclidean supergravity c-map metrics and show that they are para-quaternionic Kahler.

2016-01-19hep-thmath.DG

Compact complex surfaces with geometric structures related to split quaternions

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that a compact oriented 4-manifold carries a para-hyperkähler structure iff it has a metric of split signature together with two parallel, orthogonal and null vector fields. Every compact complex surface admiting a para-hyperhermitian structure has vanishing first Chern class and we s…

2012-07-31hep-thmath.DG

Para-Hermitian and Para-Quaternionic manifolds

A set of canonical parahermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on $Sol^4_1$. A locally …

2005-07-06hep-thmath-phmath.DG