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Results tagged with riemannian-geometry
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user 25511
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
9
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1
answer
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When do the lengths of simple closed curves determine a hyperbolic surface?
Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g …
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4
votes
2
answers
265
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Metrics with fixed conformal structure and diameter
I have three questions.
I consider a sequence of metrics $h_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h_n$ is always $1$. Is it possibl …
- 2,696
8
votes
3
answers
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Zone of negative curvature on surfaces embedded in $\mathbb{R}^3$
I consider the standard embedding of a compact oriented surface $\Sigma$ (say of genus 2) in the Euclidean space $\mathbb{R}^3$. I have coloured on the picture below the zone of this surface where the …
- 2,696
8
votes
2
answers
221
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Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
Good afternoon everyone !
I have the following question of Riemannian geometry :
Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth Riemanni …
- 2,696
10
votes
2
answers
4k
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Examples of non isometric surfaces having the same curvature function
I think it is really natural to believe, after doing Riemannian geometry for a little time, that sectional curvature encodes the all local geometry of a Riemannian manifold. One of the first thing one …
- 2,696
5
votes
2
answers
542
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Are ramified covering of negatively curved manifolds negatively curved?
Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, ca …
- 2,696
3
votes
2
answers
220
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Random metrics on compact orientable surfaces
Hello everyone,
Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of unif …
- 2,696
12
votes
2
answers
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Negative sectional curvature and constant curvature
Good morning everyone,
I was wondering about the difference between manifolds carrying a Riemannian metric with negative sectional curvature and hyperbolic manifolds. I was told once "there are very f …