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Results tagged with riemannian-geometry
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user 68969
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
380
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Perturbing metrics with nonpositive curvature
Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian …
CommunityBot
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3
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1
answer
248
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Asymptotic parametrization for negatively curved surfaces
Let $S$ be a complete simply connected negatively curved surface immersed in Euclidean space $\textbf{R}^3$. Does there exist a parametrization $f\colon\textbf{R}^2\to\textbf{R}^3$ for $S$ such that t …
- 108k
4
votes
Accepted
Can every surface be realized as a mean convex hypersurface in $\mathbb{R}^3$?
Yes, every closed orientable surface can be embedded in $R^3$ with positive mean curvature.
One way to construct higher genus examples is by gluing thin tori, which are mean convex. For instance we ca …
- 7,179
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vote
2
answers
219
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A triangle comparison in CAT(0) spaces
Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in x …
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A triangle comparison in CAT(0) spaces
This is just a bit more elaboration on Anton's nice example, and subsequent comment. Here is the picture of the two triangles:
Note that $p'x'y'$ is obtained from $pxy$ by rotating the side $\overlin …
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4
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1
answer
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Convex hull of 3 points in Cartan-Hadamard manifolds
Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the Euclide …
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3
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Accepted
Convex hull of 3 points in Cartan-Hadamard manifolds
I believe that the idea described by Ian Agol works, and can be elaborated on as follows. The general fact we want to establish is that the convex hull of a finite collection $X$ of points in $M$ is n …
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5
votes
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answer
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Manifolds with nonpositive radial curvature
How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not kn …
- 45k
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vote
0
answers
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Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property q …
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Convex surfaces with minimal total curvature in Cartan-Hadamard 3-space
The following paper develops the outline described by Anton Petrunin to solve Gromov's problem on total absolute curvature for simply connected surfaces:
Convexity and rigidity of hypersurfaces in Car …
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6
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2
answers
375
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Convex surfaces with minimal total curvature in Cartan-Hadamard 3-space
A Cartan-Hadamard 3-space $M$ is a complete simply connected 3-dimensional Riemannian manifold with nonpositive sectional curvature. A (smooth) convex surface $\Gamma\subset M$ is an embedded topologi …
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4
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1
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Approximate isometric embeddings of surfaces
The fundamental theorem of surfaces states that if symmetric matrices $g_{ij}$, $l_{ij}\colon U\subset R^2\to R$, where $U$ is open and $g_{ij}$ is positive definite satisfy the Gauss and Codazzi equa …
- 108k
11
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Riemann's formula for the metric in a normal neighborhood
I came to this post many years later, since I too was concerned about the absence of Riemann's formula in most texts, lengthy treatment in others, or reliance on more advanced techniques like Jacobi f …
- 7,179
17
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How should you explain parallel transport to undergraduates?
I agree with Ben McKay and Robert Bryant that the best way to introduce parallel transport to students, or to provide some motivation and intuition for it, is via an extrinsic approach, i.e., by the e …
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1
answer
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Converse to Euclid's fifth postulate
There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, o …
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