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**11**

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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 ...

**1**

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**2**answers

317 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 ...

**7**

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**1**answer

839 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

569 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**

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**1**answer

290 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 $Y$, ...

**18**

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**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 ...

**4**

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**0**answers

238 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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**0**answers

227 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 ...

**15**

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**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 ...