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      Questions tagged [dg.differential-geometry]

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

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      49 views

      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 ...
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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 ...
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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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      1answer
      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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      1answer
      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 ...

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