Questions tagged [quaternionic-geometry]

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9
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1answer
425 views

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 ...
1
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1answer
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 ...
2
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0answers
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 ...
1
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1answer
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 ...
8
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0answers
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 ...
2
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0answers
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 ...
4
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0answers
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 $V$ ...
3
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1answer
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 $$ g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1} $$...
12
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1answer
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-...
2
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0answers
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....
12
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0answers
170 views

Where can I find a copy of Bérard-Bergery's lecture notes on quaternionic manifolds?

In the 1970's, Bérard-Bergery proved certain results on quaternionic Kähler manifolds, some of which are explained in the book Einstein Manifolds by Besse. Several times, Besse's book references a set ...
7
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1answer
511 views

The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\...
4
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1answer
198 views

Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page. I am asking if it ...
4
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0answers
104 views

Hodge-Weil Formula for Quaternionic-Kähler manifold

Let $M$ be a quaternionic-Kähler manifold, with fundamental form $\omega$, and let $L$ be the Lefschetz operator of $\omega$. In the Kähler and, more generally, symplectic cases, there is a mysterious ...
7
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2answers
350 views

Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds

As is well-known (see Friedrich's book for example) every Kähler manifold is spin (or at least spin$^c$) and the Dirac is given (up to a twist) by $\partial + \partial^*$. What happens in the ...
4
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2answers
230 views

Are Wolf spaces flag manifolds?

It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general ...
4
votes
1answer
183 views

Homogeneous Quaternionic-Kähler Structure of the Grassmannians?

Paraphrasing from Cortes' notes: The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a parallel subbundle $Q$ which is locally spanned by $3$ ...
10
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2answers
420 views

References on quaternionic geometry

Is there any analog, in the quaternionic setting, of Kahler potentials? In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...
5
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0answers
243 views

Reference request: 3-dimensional Mobius transforms

I am working on a project that I suspect requires calculations involving Möbius transformations on 3-dimensional space $\mathbb{R}^3$, identified with the quaternions $\mathbb{H}$ with $k$-component ...
13
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3answers
411 views

Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...
2
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2answers
340 views

What turns $k$-variety into $k$-manifold?

Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A_k^n$, and that variety inherits the structure of $k$-manifold. However, the above ...
9
votes
1answer
2k views

Primes that are the sum of three squares

This is in some sense an extension of the earlier MO question, "Gaussian prime spirals." Gaussian primes in the complex plane, $a+b i$, require $a^2 +b^2$ prime off the axes. The generalization to ...
8
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2answers
1k views

Hyper-complex and quaternionic Kähler Geometry

What is the exact relationship hyper-complex and quaternionic Kahler manifolds? From Wikipedia we get that hyper-Kahler manifolds are both hyper-complex and quaternionic Kahler. Thus, the two families ...
5
votes
1answer
514 views

Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?

I am interested in quaternionic-Kahler metrics that are "as inhomogeneous as possible." Every complete quaternionic-Kahler manifold $X$ I can remember hearing of is a discrete quotient of some $Y$, ...
21
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1answer
1k views

The Quaternion Moat Problem

"One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large ...
6
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0answers
303 views

Quaternionic Veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow \mathbb{C}P^3$ is ...
1
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0answers
275 views

The geometry of PSO(4) and the quaternions [closed]

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
21
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3answers
1k views

Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds?

I start with some background, but people familiar with the subject may jump directly to the question. Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an almost hypercomplex structure ...