New answers tagged

1 vote

Isoperimetric inequality for minimal surfaces bounded by space curves containing a line segment

The area minimizing surface $\Sigma$ with boundary $\gamma$ has nonpositive curvature in the sense of Alexandrov. Applying Reshetnyak majorization theorem (see 9.56), we get a convex plane figure $F$ ...
3 votes

Coherent sheaves and holomorphic vector bundles

I think the general idea of "coherent sheaves are what you get what you try to make the category of vector bundles into an abelian category" is a good intuition to have, and this is true for ...
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0 votes

Some questions on a paper of Wilking

To answer your second question, this is just the Gromov-Hausdorff limit of $C$ with the metric rescaled by $1/\lambda,$ where $\lambda \to \infty.$ In a metric sense, $C$ is "asymptotic" to $...
2 votes

Indecomposable integral currents

I think the following might be an example, though it will require a bit of work if you want to make it more precise: Take an immersion of a sphere, which is injective except for one cap at each pole, ...
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1 vote

isoperimetric problems on Alexandrov spaces

The existence follows from theory of currents the same way as for Riemannian manifolds. As far as I know, there are no regularity results for $\partial D$. But look at the proof of Levy--Gromov ...
1 vote

Can a manifold have a curvature-free connection that is not torsion-free?

I'm surprised that nobody mentioned the Tanaka-Webster connection of a strictly pseudoconvex CR spherical manifold yet. A strictly pseudoconvex pseudo-Hermitian manifold is a triplet $(M,\theta,J)$ ...
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4 votes

Bounds for metric in normal coordinate

APOLOGIES: Major revision, including details, below. My original answer was incorrect. Thanks to @IgorKhavkine for pointing out that the question asks for a point wise bound on only the metric itself ...
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6 votes

Bounds for metric in normal coordinate

As mentioned by Deane Yang in the comments and his (deleted) answer, one can estimate the components of the metric in normal coordinates using a transport ODE (I know it from Dolgov-Khriplovich (1983) ...
4 votes
Accepted

What are some explicit examples of nontrivial gradient almost Ricci solitons with harmonic curvature?

I'm revising my answer to shorten it, since there is a much simpler way to describe these solutions more fully. Let $(N^n,h)$ be a metric of constant sectional curvature $k$ and consider the quadratic ...
7 votes
Accepted

Initially horizontal geodesic is always horizontal

After choosing local coordinates, by the implicit function theorem (I'm omitting a bunch of technical computations) there is a smooth function $\varphi:TE \to TE$ such that $\varphi(x,-): T_x E \to ...
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5 votes

Maximum symmetry metric on $ \mathbb{C}P^n $

I just wanted to add two points: A bi-invariant metric on a compact Lie group $G$ does not always induced the maximum symmetric metric on $G/H$. The most familiar examples are spheres: $S^{2n+1} = ...
  • 6,021
7 votes

Complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles

There are many such examples, for instance, all complex nilmanifolds (and most complex solvmanifolds) have tangent bundle which is topologically trivial. The Hopf manifolds also have topologically ...
4 votes
Accepted

Length and curvature for closed curves in negatively curved spaces

The Reshetnyak majorization theorem (see 9.56) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be majorized by a convex plane figure $F$; that is, there is short (= ...
1 vote

Cone unfolding of space curves

Pardon me for this bit of self-promotion, especially because this is only tangential to the OP's concerns. But cone unrolling and Anton Petrunin's mention of Alexandrov's developments, in conjunction ...
1 vote

Cone unfolding of space curves

Liberman used cylinder unfolding to study geodesics on convex surfaces. [Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313.] Right now standard ...
3 votes

For $f$ geodesically convex with $L$-Lipschitz-gradient on hyperbolic space, is $f(x)-f(x^*)\leq(\mathrm{const}) \cdot L r$ for all $x \in B(x^*, r)$?

The function $f$ has to be Lipschitz. Assume contrary, choose a sequence of points $p_n$ such that $\lambda_n=|\nabla_{p_n}f|\to \infty$. Let $$f_n(x)=\tfrac1{\lambda_n}\cdot f\circ m_n(x)-f(p_n)$$ ...
4 votes
Accepted

Upper bound on volume growth of area minimizers

Consider the complex curve $w = z^k$ in $\mathbb{C}^2\simeq\mathbb{R}^4$, which is calibrated and therefore area-minimizing. The area of the part of this curve that lies inside the polydisk $\max\{|z|...
15 votes
Accepted

Is an inextensible manifold necessarily compact?

Yes, $M$ must be compact. In fact, if $M$ is non-compact, it admits a non-surjective self embedding $f:M\rightarrow M$. When $n=1$, the only non-compact manifold is $\mathbb{R}$, which obviously ...
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4 votes

Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

This result is stated as Exercise 7 in Chapter 8 of Bröcker and Jänich's book Introduction to Differential Topology (p. 86 of the English translation). This may or may not count as a citeable ...
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6 votes

Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

The result in the following paper implies that open star-shaped domainin $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. But, in your case, a diffeomorphism can be obtained along the same lines. K. ...
2 votes

Comparing diffusion processes in different metrics

The generator of the diffusion corresponding to a Riemannian metric (i.e., the diffusion process which the limit of the random walks such that their increments go along geodesics and the ...
3 votes

A question on a result of Colin de Verdière

Vedrin Šahović in his unpublished thesis [Approximations of Riemannian manifolds with linear curvature constraints, 2009] proved that any compact metric space can be appoximated by hyperbolic ...
16 votes

Topology of the space of embedded genus $g$ surfaces in $S^3$

The first observation is that $\mathcal{E}_g$ is not connected when $g > 0$. This is due to the existence of "knotting". For example, in genus one, let $K$ and $K'$ be smooth knots. ...
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2 votes
Accepted

Intersection of conical neighbourhoods on a polyhedral space

You say "The same is true for a tubular neighborhood of the edge". This is not correct, but it is true if you stay away from the endpoints. So $U$ should be defined as a tubular neighborhood ...
0 votes

Minimize the area of a maximal surface in R^{2,1} with boundary on the unit sphere

That's a very interesting question but I don't think that the estimate as it is proposed can hold (as suggested also in the update at the end of the question). (We just discussed this with Nathaniel ...
2 votes

Is it possible to average a riemannian metric over an action and preserve curvature bounds?

As it was mentioned by Igor Belegradek, the curvature gets destroyed by averaging. Assume there is a modified the averaging process that preserves positive curvature. Then you would prove Hopf ...
1 vote
Accepted

Continuity of a reaching time of a submanifold

If $V$ is closed in $\mathcal{O}$, then $\tau^V$ is continuous at $x_0$. If $V$ is not closed, then it is not hard to find counterexamples (e.g. imagine $\mathcal{O}=\mathbb{R}^3$, $V$ is an open disk ...
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1 vote

trapped geodesics

Let us shoot a geodesic from a random point in a random direction. Note that with probability 0 it is one-sides finite + other-side infinite. However, if you shoot a geodesic from a nonregular point, ...
2 votes

A characterization of round sphere

Suppose $\gamma$ is a geodesic in $M$. Then there is a function $\ell$ such that $|\gamma(t_0)-\gamma(t_1)|=\ell(|t_0-t_1|)$, assuming that $|t_0-t_1|$ is small. It follows that $\gamma$ has constant ...
4 votes
Accepted

Smooth mapping from $\mathrm{RP}^2$ to $\mathbb{R}^3$ with nonsingular derivative

This is a famous problem that was solved by a doctoral student of David Hilbert named Werner Boy in 1901. The kind of mapping you are looking for is called an "immersion" (of its domain — ...
4 votes
Accepted

Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy

For torus knots, all of the representations into $SU(2)$ were rather explicitly worked out by Eric Klassen (Representations of knot groups in $SU(2)$. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–...
3 votes

Usefulness of Nash embedding theorem

Let me add that the generalized Gauss–Bonnet formula was first proved for embedded Riemannian manifolds. It was done independently in Weyl's "On the volume of tubes" (1939) and Allendoerfer'...
6 votes
Accepted

Space of spacelike embeddings as infinite-dimensional manifold

A standard reference on infinite dimensional manifolds is Kriegl, Andreas; Michor, Peter W., The convenient setting of global analysis, Mathematical Surveys and Monographs. 53. Providence, RI: ...
2 votes

Integration and Stokes' theorem for vector bundle-valued differential forms?

I think we have the (integration over the fiber) results you're looking for in our paper Let me state our proposition word for word here just to give a taste: Proposition 5.1 Let $F \hookrightarrow Y \...
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7 votes

Making a submanifold transverse to a vector field by an isotopy

The simplest case is when $M$ is a compact manifold with connected boundary $N$. If $N$ is nowhere tangent to $X$ then, by replacing $X$ by $-X$ if necessary, we can assume $X$ points outwards at ...
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0 votes

From time-dependent Hamiltonians to time-dependent symplectic/Poisson structures

Let's denote $H_t$ for the Hamiltonian function. The support of $X_t^{\omega_t}$ is the support of $dH_t$ independent of the symplectic form $\omega_t$. So if for example the interior of the support ...
10 votes
Accepted

Is every retraction homotopic to a smooth retraction?

Using a collar of the boundary we resort to the case when $r\in C^0(M, \partial M)$ is given by the projection $\partial M\times I \to \partial M$ over the collar . Since $r$ is smooth on an open ...
13 votes
Accepted

Maximum symmetry metric on $ \mathbb{C}P^n $

There's an easy counterexample to your guess: Let $M^6 = \mathrm{SU}(3)/\mathbb{T}^2$, where $\mathbb{T}^2\subset\mathrm{SU}(3)$ is the maximal torus (for example, the diagonal subgroup). In that ...

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