Linked Questions
11 questions linked to/from Can every manifold be given an analytic structure?
44
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8
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What's the difference between a real manifold and a smooth variety?
I am teaching a course on Riemann Surfaces next term, and would like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular ...
15
votes
2
answers
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What is known about the MMP over non-algebraically closed fields
I would like to know how much of the recent results on the MMP (due to Hacon, McKernan, Birkar, Cascini, Siu,...) which are usually only stated for varieties over the complex numbers, extend to ...
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9
votes
1
answer
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Is a manifold generically real analytic (with generic real analytic metric)?
I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
- 123
16
votes
3
answers
1k
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Analog of Newlander–Nirenberg theorem for real analytic manifolds
It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...
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12
votes
2
answers
978
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Are conformal maps between Riemannian manifolds real-analytic?
This is a cross-post.
Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map.
Do there exist ...
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9
votes
0
answers
998
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Complexification of smooth manifolds
Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?
By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
- 435
7
votes
1
answer
632
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Smooth and analytic structures on low dimensional euclidian spaces
So it is relatively easy to show that there exists only one smooth structure on
the real line $\mathbb{R}$. So here are 2 natural questions:
Q1: Up to equivalence, is there only one real analytic ...
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7
votes
0
answers
775
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How much differs the category of real-analytic manifolds from $C^\infty$ ones?
I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
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7
votes
0
answers
499
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Smoothing a piecewise smooth manifold
Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can ...
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13
votes
0
answers
364
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What is known about differentiable and analytic structures on the long line (and half-line)?
When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ ...
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4
votes
0
answers
121
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planar linkage isomorphic to an exotic sphere
I recently came across this paper, which showed that any compact smooth manifold is diffeomorphic to a connected component of the moduli space of a planar linkage.
Briefly, if we have an undirected ...
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