# Questions tagged [quaternionic-geometry]

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### What is the convex hull of the quaternionic symmetries of the 3 dimensional cube?

It is well known that there are exactly five 3-dimensional regular convex polyhedra, known as the Platonic solids. In 1852 the Swiss mathematician Ludwig Schlafli found that there are exactly six ...
202 views

### compact manifold as a hyperkahler quotient of an infinite-dimensional affine space

Is it possible to obtain K3 (or any other compact hyperkahler manifold) with its hyperkahler structure as a hyperkahler quotient of an infinite-dimensional affine quaternionic vector space with an ...
116 views

### Holonomy of hypercomplex manifold

The following is a quote from M. Barberis, I. Dotti, M. Verbitsky, Canonical Boundles of Complex Nilmanifolds with Applications to Hypercomplex Geometry, Math. Res. Lett., 16(2), 331-347, 2009. "Not ...
171 views

### Do these definitions of integrable quaternionic structure agree?

I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another. One definition that I have found (from Differential ...
85 views

### Is there a quaternionic analogue of Kodaira's embedding theorem?

Let $M$ be a $4m$-dimensional Quaternion-Kähler manifold of positive scalar curvature. Does there exist an $n$ large enough, so that $M$ can be embedded inside $\mathbb{H}P^n$ via a quaternionic ...
55 views

### What are the Cartan geometries modeled on $\mathbb{H}P^m$?

I am not an expert on Cartan Geometry (in fact, I have just read and understood the definition, at a basic level). I have the following questions: 1) Can someone please describe what are the Cartan ...
66 views

### Do geodesics on a hyperkähler quotient have nice lifts?

Suppose one has a flat quaternionic vector space $V$, with a compatible inner product $g$. So $(V,g)$ is a flat hyperkähler manifold. Assume that there is some compact Lie group $G$ acting on $V$ ...
188 views

### How does the concept of a hermitian metric generalize to a hyperkahler manifold?

A complex manifold admits an almost complex structure, $J$, which satisfies $${J^i}_j{J^j}_k=-{\delta^i}_k,$$ and a Hermitian metric, $g$, which satisfies $$g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1}$$...
276 views

### Compact quaternionic Kahler manifolds of negative curvature: examples

There is a well known problem of LeBrun-Salamon: are there any non-symmetric compact quaternionic-Kahler manifolds of positive scalar (and Ricci) curvature? It is hard and still unsolved: Quaternionic-...
129 views

### Terminology: Almost hyper-Hermitian vs Almost hyper-Kähler vs

After non-thorough literature search, it seems to me that there is no consensus on the usage of the terminology "(almost) hyper-Hermitian" vs "almost hyper-Kähler" vs "almost quaternion-Hermitian" etc....