# Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

5,746 questions
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### Compact complex affine Kähler manifold is a torus

Before giving a motivation let me ask the precise question firstly. By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which ...
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### Differential rotations in Chebyshev net

A Chebyshev net obeying Sine-Gordon equation is drawn on a surface of constant negative Gauss curvature $K$ so that the asymptotic differential rhombic element corners lie on lines of maximum/minimum ...
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### Geometric bang-bang theorem for nonlinear optimal control

The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
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### Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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### $(M,g)$ is complete iff $(\tilde{M},\tilde{g})$ is complete (non-Riemannian version)

I'm not sure if this question is too low level for Math Overflow (so feel free to move this to SE if you think it is). Inspired by this and this question I'm wondering if the following statement is ...
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### harmonic coordinates on non-compact manifolds

Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
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### Diffeological spaces and Sikorski differential spaces

Diffeological spaces and Sikorski differential spaces are each a generalisation of a smooth manifold. In their definitions, both have locality and smooth compatibility conditions. Any diffeological ...
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### Using Stokes' theorem to define “area” enclosed by a curve

I am trying to figure out what the next calculation of the "area" (or "volume" in higher dimensional analogues) using Stokes' theorem really means. Here is my thought process: $2$-dimensional case: ...
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### Bieberbach theorem for compact, flat Riemannian orbifolds

In his thesis, Bieberbach solved Hilbert 18 problem and proved that any compact, flat Riemannian manifold is a quotient of a torus. I need a reference to an orbifold version of this result: any ...
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### Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...
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### Rolling without slipping interpretation of torsion

This is, in a sense, a follow up to this question. Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...
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### Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?

Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
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### An introduction paper or book to Spectral Flow

Is there someone can tell me some papers or books about the basic material of Spectral flow? I want to know, what is spectral flow and how to use it to geometry.
I would like to ask if there is a holomorphic version of Darboux's theorem. More concretely, given a holomorphic symplectic manifold $(X, \omega)$ is there a local holomorphic symplectomorphism from $(... 1answer 208 views ### Is every basis for$\bigwedge^kV$satisfying a “complementary” property a rescaling of a “standard” basis? This is a cross-post. Let$V$be a$4$-dimensional real vector space. Let$\omega_{i_1,i_2}$be a basis for$\bigwedge^2V$, where each$\omega_{i_1,i_2}$is decomposable. Suppose that for every$\...
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is ...