In this article we discuss the interaction between the geometry of a quaternion-Kahler manifold M and that of the Grassmannian G(3,g) of oriented 3-dimensional subspaces of a compact Lie algebra g. This interplay is described mainly through the moment mapping induced by the action of a group G of quaternionic isometries on M. We give an alternative expression for the endomorphisms I_{1},I_{2},I_{3}, both in terms of the holonomy representation of M and the structure of the Grassmannian's tangent space. A correspondence between the solutions of respective twistor-type equations on M and G(3,g) is provided.
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.
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.
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes benefit of the quaternionic structure and the results are much simpler than the present formulations of quaternionic curves.
Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient o…
We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We show that the quaternionic Monge-Ampère equation on these manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.
Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry
超复流形是具有满足四元数关系的复结构$I, J, K$的三元组的流形。定义了超复流形上多重次调和函数的四元数模拟，并将其几何解释为HKT（hyperkähler with torsion）度量的势，证明了A. D. Aleksandrov定理和Chern-Levine-Nirenberg定理的四元数模拟.
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…
We show that the geometry of $4n$-dimensional quaternionic Kähler spaces with a locally free $\mathbb{R}^{n+1}$-action admits a Gibbons-Hawking-like description based on the Galicki-Lawson notion of quaternionic Kähler moment map. This generalizes to higher dimensions a four-dimensional construction, due to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local c-map proposed by Roček, Vafa and Vando…
