Questions tagged [quaternionic-geometry]
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36
questions
27
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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
...
- 149k
24
votes
3
answers
2k
views
Are there topological obstructions to the existence of almost quaternionic structures on compact manifolds?
$\DeclareMathOperator\End{End}\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$I start with some background, but people familiar with the subject may jump directly to ...
- 2,151
13
votes
3
answers
480
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 ...
- 2,438
12
votes
1
answer
395
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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-...
- 9,095
12
votes
0
answers
185
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 ...
- 121
10
votes
1
answer
532
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 ...
- 2,233
10
votes
2
answers
481
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 ...
- 101
9
votes
1
answer
3k
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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 ...
- 149k
8
votes
2
answers
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 ...
- 3,359
8
votes
1
answer
581
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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 {\...
- 347
8
votes
0
answers
92
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 ...
- 5,011
7
votes
2
answers
495
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 ...
- 1,383
6
votes
0
answers
313
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 ...
- 101
5
votes
1
answer
595
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 $Y$, ...
- 1,168
5
votes
1
answer
228
views
Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure
Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
- 91
5
votes
0
answers
399
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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 ...
4
votes
1
answer
208
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 ...
- 347
4
votes
2
answers
245
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 ...
- 1,383
4
votes
1
answer
225
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 $3$
...
- 181
4
votes
0
answers
158
views
Canonical bundle of twistor space
I am wondering whether there are cases, and if so whether some sufficient assumptions are know, when the twistor space of compact quaternionic/hypercomplex manifold has trivial canonical bundle. As ...
- 437
4
votes
0
answers
73
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 $V$ ...
- 5,011
4
votes
0
answers
223
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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....
- 1,337
4
votes
0
answers
112
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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 ...
- 465
3
votes
1
answer
223
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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
$$
g_{st}{J^s}_i{J^t}_j=g_{ij}. \tag{1}
$$...
- 1,135
3
votes
0
answers
110
views
Which non-compact quaternion-Kähler spaces are Kähler?
The list of quaternion-Kähler compact symmetric spaces can be found here. I am curious to know which of the non-compact versions of these spaces are Kähler. If the answer is known also for non-...
- 31
2
votes
2
answers
352
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 ...
- 2,175
2
votes
1
answer
221
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What do the Pauli matrices say about the Threefold Way?
The Pauli matrices
$$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},
\sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},
\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 & -1 \...
2
votes
0
answers
72
views
Bézout theorem for complex solutions of systems of homogeneous quaternionic polynomials
Suppose that I have a system of $n$ homogeneous polynomials in $n+1$ variables $p_{1},\dots,p_{n}\in\mathbb{H}[x_{0},\dots,x_{n}]$ with coefficients at both sides in $\mathbb{H}.$ I want to know if ...
- 573
2
votes
0
answers
154
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 ...
- 437
2
votes
0
answers
83
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 ...
- 5,011
1
vote
1
answer
207
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 ...
- 1,135
1
vote
1
answer
280
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 ...
- 9,095
1
vote
0
answers
438
views
Heat kernel on quaternion Heisenberg group
For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
- 11
1
vote
0
answers
120
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Conway's quaternion notation +1/3[C_3×C_3]∙2^(2) represent C_3h of 4d Point group?
In John Conway and Derek Smith's On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry, they introduce a way to connect quaternions to 4D Point Group.
Suppose: $[l,r]:x\to \bar lxr\;,\...
- 11
1
vote
0
answers
147
views
Fundamental $1$-form for a Riemannian manifold?
Take a Hermitian manifold $(M,I,g)$ where $I$ is the complex structure and $g$ is the Hermitian metric. The associated fundamental $2$-form
$
g(\cdot,I(\cdot))
$
captures a lot of the information ...
- 203
1
vote
0
answers
285
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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 ...
- 106