The quaternionic-geometry tag has no usage guidance.

**6**

votes

**1**answer

355 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**

votes

**1**answer

171 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 ...

**1**

vote

**0**answers

47 views

### Hypercomplex, hyperKahler, or quaternion-Kähler from Joins/Connected Sums

I am looking for examples of (compact) hypercomplex, hyperKahler, or quaternion-Kähler manifolds which can be constructed as joins/connected sums of manifolds which do are not hypercomplex, ...

**4**

votes

**0**answers

84 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 ...

**5**

votes

**2**answers

182 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**

votes

**2**answers

151 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

**1**answer

103 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**

votes

**2**answers

332 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**

votes

**0**answers

110 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**

votes

**3**answers

346 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**

votes

**2**answers

321 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

**1**answer

1k 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 ...

**5**

votes

**2**answers

736 views

### Hyper-Complex and quaternionic Kahler 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

**1**answer

373 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 $Y$, ...

**19**

votes

**1**answer

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 ...

**5**

votes

**0**answers

267 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**

vote

**0**answers

243 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 ...

**18**

votes

**3**answers

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 ...