Questions tagged [geometric-structures]

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What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...
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3 votes
0 answers
144 views

Failure of the Jacobi identity

So I'm facing a problem of physical origin which I'd like to understand on a geometric background. I have a long, tedious bivector involving functional derivatives. I write what it would be the ...
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5 votes
1 answer
205 views

Why is generalised complex structure defined to be a reduction of structure group to $O(n,n) \cap Gl(n,\mathbb{C})$?

It is a basic and "intuition request" question. I have asked it on StackExchange yet it is probably to specialized for it since there were no answears. Generalised complex structure is defined to be ...
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2 votes
1 answer
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Gromov Geometric Structures and Killing fields

Let's fix some notations: $M$ will denote a real smooth, $m$-dimensional, manifold, $F^k(M)$ is the k-th order frame bundle on $M$ and $Gl^k(m)$ is the space of $k$-jets of diffeomorphisms of $\mathbb ...
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5 votes
2 answers
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What criteria are there to determine if two projective varieties are projectively equivalent?

A projective transformation is a morphism of $P^n$ to $P^n$, for some $n$, determined by an $(n + 1) \times (n + 1)$ invertible matrix $A$ in the obvious way. The sets $Q$, $R$ are projectively ...
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10 votes
1 answer
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necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds

Is there any necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds $M$?
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35 votes
3 answers
2k views

"Softness" vs "rigidity" in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually,...
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0 votes
1 answer
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Non Smooth K3 surface?

Hi, My question is related to Algebraic Surfaces. I have seen that we always consider K3 surfaces which are smooth, but I wonder how can we define a non-smooth K3 surface. The problem I see is on ...
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2 votes
2 answers
230 views

Of what kind of complemented bounded poset are the structures in my quasi-variety?

I feel that my question is very basic, but, somewhat suprisingly, nobody was able to give me an answer so far: Let $\mathbf{M} := \langle \{ 0,1 \}, 0, 1, \leq, \neg \rangle$ be the structure with ...
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9 votes
5 answers
1k views

Möbius and projective 3-manifolds

A projective 3-manifold is a smooth manifold that admits an atlas with values in the real projective 3-space such that all transition maps are restrictions of projective transformations. A Möbius 3-...
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