All Questions
Tagged with metric-spaces riemannian-geometry
250
questions
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Confusion about the criteria for a complete metric space to be a length space
According to corollary 2.4.17 from the textbook "A Course In Metric Geometry" we have
A complete metric space $(X, d)$ is a length space iff,
given a positive $\varepsilon$ and two points $x,...
- 65
3
votes
1
answer
54
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Accessibility Components does *not* coincide with Path Connected and Connected Components for Length Spaces
I am working through "A Course in Metric Geometry" and I do not believe this exercise to be true:
Exercise $\boldsymbol{2.1.3}$
$3)$ Verify that accessibility components coincide with both ...
- 65
1
vote
1
answer
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Are Metric Balls on a Riemannian Manifold Embedded Submanifolds?
Let $(M,g)$ be a connected $n$-dimensional Riemannian manifold. We can define a topological metric on $M$ by
$$ d_g(p,q) = \inf\{\text{Length}(\gamma) \mid \gamma \ \text{is a piecewise smooth curve ...
- 515
1
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1
answer
57
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Complete Riemann Manifold
I was trying to understand this article about the existence of complete Riemannian metrics by Nomizu Ozeki, see The Existence of Complete Riemannian Metrics, Proceedings of the American Mathematical ...
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Probability measure satisfying curvature-dimension
Assume that the non-compact $n$-dimensional Riemannian manifold $M$ satisfies a curvature-dimension $\mathrm{CD}(K,N)$, $K \le 0$, with respect to a measure $\mu = e^{-V} d\mathrm{Vol}$; i.e.
\begin{...
- 397
1
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54
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Metric choice in the tangent space of a Riemannian manifold obtained through the Log map
After using the Log map, as defined in this paper Riemannian approaches in Brain-Computer Interfaces: a
review (Section III. A page 2&3), to project points from the manifold onto the tangent space ...
- 51
2
votes
1
answer
134
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What is the reason for endowing the positive semidefinite cone with the Affine Invariant Riemannian Metric instead of a Euclidean metric?
I came across this post Distances defined in manifold of symmetric positive definite matrices because I had the same questions. I did not understand the answers provided, so I wanted to try to answer ...
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25
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Length of rectifiable curves in Finsler spaces
Let $U$ be an open set in $\mathbb{R}^n$, let $E$ be the set of norms on $\mathbb{R}^n$, and let $N: U\rightarrow E$ be a map such that $(x,v) \mapsto N(x)(v)$ is continuous.
We define the length of a ...
- 2,719
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41
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Mixing metric tensors on Riemannian manifold
I am given a smooth connected manifold $M$ with two smooth metric tensors $g_1$ and $g_2$. Let us denote by $d_1$ and $d_2$ the distance they induce on $M$, respectively. I have two ways of "...
- 1,456
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86
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Statistical invariants of Riemannian Manifolds
A cheap way of defining invariants of Riemannian manifolds?
Let $(M, g)$ be a Riemannian Manifold with finite volume and $X=(X_1,\dots, X_n)\sim \mathscr U(M^n)$ a random sample with respect to the ...
- 943
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129
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Difference between a metric sphere and a topological sphere
Recently I encountered a mysterious term named "metric $2$-sphere". Does it have anything to do with a metric space? If it is typically used to refer to a sphere in any metric space, why ...
- 175
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95
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Are all squared metric distances also divergences?
Let $M$ be a differentiable manifold that is also a metric space $(M,d)$, equipped with the topology induced by the metric distance $d$; further, assume that $d$ is $C^2$ on $M\times M$. Now, let $D:M\...
1
vote
0
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60
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Riemannian metric on fixed rank manifold
I know that one can define metrics on the manifold of SPD matrices
$$
\mathcal{S}^n = \{ A \in \mathbb{R}^{n\times n} \ | \ \text{A positive semi-definite} \}
$$
like the Log-Euclidean metric or the ...
- 11
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34
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Indefinite geodesics and infinte path
I read the theorem 1.4.7. in "Riemannian Geometry and Geometric Analysis" (2005) that states:
$\textbf{Theorem 1.4.7.}$ Let $M$ be a compact Riemannian manifold. Then for any $p \in M$, the ...
4
votes
1
answer
58
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Minimal length in the free homotopy class as translation length
Let $M$ be a compact Riemannian manifold. The fundamental group $\pi_1(M,p_0)$ is isomorphic to the group of deck transformations of the universal cover $\pi:\widetilde{M}\to M$. The translation ...
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