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Results tagged with riemannian-geometry
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user 1540
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.
2
votes
Accepted
Relationship between spectrum geometry and almost-isometry
The above comments are still mostly valid with $\epsilon$-isometries:
if $(M,g)$ and $(N,g)$ are Riemannian manifolds with diameters less than $D$, then
$f:M\to N$, $x\mapsto n_0$ for some $n_0\in N …
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9
votes
Accepted
Distance between two geodesics originating from separate but nearby points
Let $\sigma(s)$ denote a unit speed minimizing geodesic from $x$ to $y$. Let $l=d(x,y)$ be the length of $\sigma$.
Define a vector field $v(s)$ along $\sigma$ by parallel transport. Then set
$$
\Gamma …
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10
votes
Accepted
co-dimension one minimizing verifolds
I think that the phenomena is not stable under perturbation of the metric, i.e., a small perturbation can cause the minimizer to be smooth.
First, let me explain how it is not stable under perturba …
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3
votes
Accepted
The limit of a sequence of embedded minimal disks in $\mathbb{R}^3$
A first answer is (under much weaker hypothesis than you require)
Answer 1: If $\Sigma_k$ are a sequence of minimal surfaces with $\partial\Sigma_k \subset B(0,r_k)$ with $r_k\to\infty$ and $ …
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5
votes
Accepted
A surface on which all regular curves have nowhere vanishing curvature
I'm going to assume that you mean the following property
$S \subset \mathbb{R}^3$ has property (*) if for any regular curve $\gamma\subset S$, the curvature vector $\vec{\kappa}$ of $\gamma$ as a …
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6
votes
Accepted
Singularities in minimal surfaces
On one hand, the answer to the question
Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of …
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6
votes
Accepted
A Scalar Curvature Computation in Brendle Marques Neves' Min-Oo Conjecture paper
Willie Wong gives the correct computational method of computing the desired formula. If you happen to believe the general formula for the derivative of the scalar curvature $R$, you can save yourself …
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5
votes
Accepted
Taylor expansion of the square of the distance function on a Riemannian manifold
The "standard" proof using Jacobi fields can be found in section 1.3 here https://www2.math.upenn.edu/~wziller/math660/TopogonovTheorem-Myer.pdf
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3
votes
Computations with the distance function on a Riemannian manifold
UPDATE: I notice that the question leaves the possibility of boundary open. The answer below only works as written if there is no boundary. I'm not sure what happens if there is boundary, but one shou …
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12
votes
2
answers
1k
views
Converse to Bishop-Gromov Inequality
Is the converse to the Bishop-Gromov Inequality true?
In other words, if, for a complete $n$-dimensional Riemannian manifold $M$, there is $k \in \mathbb{R}$, such that defining $V_k(R)$ to be the s …
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6
votes
1
answer
388
views
Do manifolds with no Ricci lower bounds for any metric exist?
Is there a smooth (noncompact) manifold $M$ such that for any Riemannian metric $g$ on $M$ there are $p_i$ and unit tangent vectors $v_i \in T_{p_i}M$ such that $Ricc(g)|_{p_i}(v_i,v_i) \leq -i$? This …
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5
votes
Convex embedding with a positivity condition
The answer is yes. Even without convexity, a version of Alexandrov's theorem holds true for symmetric functions of the principal curvatures, i.e. if the $k$-th symmetric polynomial of the principal cu …
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1
vote
Normal variation of embedded surfaces
You may end up reading a lot more than just what you wanted to know originally, but I think that Gromov's article Sign and geometric meaning of curvature is an amazing place to look for this sort of i …
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3
votes
Complete stable minimal hypersurface in positively curved manifolds
You can construct a positively curved $(M^n,g)$ for any $n\geq 4$ that admits a stable minimal hypersurface. This is described in Example 1.2 here. (That paper also contains some non-existence results …
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4
votes
Accepted
On the Geroch's argument
I think that I misunderstood the notation in Question 1. The bold statement is incorrect. Consider the function $\tau(x) = 2x$ on $\mathbb{R}$ and the function $k(x) : = x$ on $\mathbb{R}$. Then $S_t …
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