# Questions tagged [dg.differential-geometry]

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

6,062
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### A geometric property of certain Lie groups

What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$ but satisfy the following:
For every left invariant metric $g$ we draw all geodesics. Then we CAN rescale $...

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78 views

### What is a 0-Manifold? [on hold]

The definition of $n-$manifold involves the euclidian space $\mathbb{R}^n$, so a $0-$manifold is essentially a singleton? Or just the empty set?

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90 views

### directional derivative along geodesic flow of vector field

A rather elementary question for the differential geometers. Let $M$ be a Riemannian manifold, let $X\colon M \to TM$ be a vector field, and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" of the ...

**4**

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52 views

### Effect of the inverse exponential map on the curvature of a given curve

Suppose you have a curve $\alpha$ in a manifold $\mathcal{M}$. You are at a point $\alpha(t)$ of that curve. The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}...

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60 views

### Separating two submanifolds [on hold]

Let $M, N, X$ are compact manifolds. Let $f_1:M \rightarrow X$ and $g_1: N \rightarrow X$ be any two embeddings. Is it always possible to find embeddings $f_2 $ homotopic to $f_1$ and $g_2$ homotopic ...

**2**

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66 views

### How to check conditions for Liouville-Arnold theorem?

Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...

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42 views

### Bound on the distance from points to the boundary of a hyperbolic surface

Fix $\epsilon\in\mathbb{R}_{>0}$, $\Sigma$ a surface with boundary and let $\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$ denote the Teichmüller space of hyperbolic structures of $\Sigma$ with geodesic ...

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31 views

### Submatrix of uniform distribution on Stiefel manifold

Let $U\in O(n,r)$ be uniformly distributed on the Stiefel manifold. Let
$$X=\begin{pmatrix}
U_{11}^2 & \cdots & U_{1r}^2\\
\vdots & \ddots & \vdots\\
U_{r1}^2 & \cdots & U_{rr}...

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59 views

### Volume form induces Borel measure: proof verification [migrated]

Disclaimer: I asked this question already on the regular mathematics site here, but to no avail, even with a bounty. I think answering said question is still of value.
Proposition. Let $M$ be a ...

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60 views

### Marginal surfaces in spacetimes

Is there some result on existence of marginally trapped surfaces in spacetime 4-manifolds?
Am I right in saying that a marginal surface (like a trapped surface in general) is a compact spacelike 2-...

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60 views

### Star product on functions of a Poisson-Lie group

Consider a Poisson-Lie group $G$, with whatever additional requirements (quasi-triangular, compact, simply connected).
We can consider $G$ as a Poisson Manifold and apply Kontsevich formality to ...

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109 views

### On the diameter of left-invariant sub-Riemannian structures on a compact Lie group

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...

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101 views

### Is there a family of Riemannian manifolds with explicitly solvable geodesics for manifold M?

By this paper https://www.mat.univie.ac.at/~michor/rie-met.pdf for any given manifold $M$, there is a Hilbert manifold (I believe its a Hilbert manifold, if not it is still an infinite dimensional ...

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**1**answer

88 views

### A non-geodesible foliation of $S^3$ or $S^2\times S^1$

Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...

**2**

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**1**answer

91 views

### Coinvariant representative of homogeneous space cohomology

Given a compact homogeneous space $M = K/L$, consider its de Rham complex $(\Omega^*,d)$. Will every cohomology class $[\omega] \in ker(d)/im(d)$ contain a representative $\nu$ which is invariant with ...