The quaternionic-geometry tag has no usage guidance.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $3$
...

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321 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 ...

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### 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 ...

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### 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 ...

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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 ...

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**1**answer

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### 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 ...

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### 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 ...

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### 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 $Y$, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...