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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.
11
votes
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 …
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5
votes
1
answer
329
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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 …
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1
vote
2
answers
219
views
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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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 …
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8
votes
The geometry of Nadirashvili's complete, bounded, negative curvature surface
The conjecture attributed to Hadamard, if one regards that as being concerned with the existence of a complete embedded negatively curved surface in a ball, is still open. I have read the correspondin …
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3
votes
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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9
votes
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 …
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1
vote
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
votes
2
answers
375
views
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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1
vote
0
answers
103
views
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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3
votes
1
answer
248
views
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 …
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4
votes
Is Gauss map of a free boundary convex disk a diffeomorphism?
The answer is yes. To show this one can use the fact that any topological immersion (locally one-to-one continuous map) of an n-dimensional disk into a sphere of the same dimension is an embedding (gl …
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4
votes
1
answer
190
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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 …
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4
votes
1
answer
175
views
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
votes
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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