3
votes
2answers
295 views

Reference request for an early theorem of Gromov

In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter ...
12
votes
4answers
738 views

What is the analog of the “Fundamental Theorem of Space Curves,” for surfaces, and beyond?

The "Fundamental Theorem of Space Curves" (Wikipedia link; MathWorld link) states that there is a unique (up to congruence) curve in space that simultaneously realizes given continuous curvature ...
1
vote
0answers
278 views

Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...
11
votes
3answers
241 views

Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us: Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...
3
votes
0answers
73 views

The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
3
votes
1answer
259 views

Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic? For example, is this true for small ...
4
votes
3answers
472 views

Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below : This post has been divided into two parts, the second part is here. Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
1
vote
3answers
168 views

Geometric means of matrices beyond the positive definite cone

Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for ...
0
votes
0answers
59 views

Minkowski functional on the sphere

Let $\mathbb{S}$ be the unit sphere, and let $v_1\in\mathbb{R}^d$ be the north-pole. Let $\mathcal{E}$ be a symmetric convex body (with respect to the geodesics on the sphere) and let the minkowski ...
4
votes
0answers
224 views

Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements $\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...
6
votes
1answer
509 views

How metric is Riemannian geometry

Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by $$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, ...
4
votes
0answers
115 views

Gromov-Haussdorf and Lipschitz convergence of a non-collapsing sequence of manifolds with Ricci curvature bounded below

There is a theorem from Cheeger-Colding saying the following: Let $n$ be an integer. If a sequence of $n$-dimensional Riemannian manifolds $(M_i,g_i)$ converges with respect to the Gromov-Hausdorff ...
7
votes
1answer
438 views

complete metric space

Hallo, I have the following question: Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: ...
4
votes
1answer
230 views

Alexandrov angles in Riemannian manifolds

Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch ...
9
votes
1answer
197 views

Positively curved manifold with almost extreme diameter

Suppose $M$ is a 1-connected closed manifold with sectional curvature $\ge 1$. So the diameter $D$ of $M$ satisfies $$ D \le \pi $$ When equality holds $M$ is isometric to round sphere. In fact this ...
3
votes
1answer
139 views

Is geodesic plane field a Killing field?

Let $M$ be a closed orientable Riemannian manifold. Recall that a plane field on a Riemannian manifold is said to be geodesic if any geodesic tangent to the plane field at one point is tangent to it ...
1
vote
1answer
73 views

3-dim 1-connected Alexandrov manifold with curvature $\ge 0$ Heomomorphic to sphere?

For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold. Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower ...
3
votes
1answer
147 views

Dose closed Alexandrov space admit a bi-Lipschitz embedding into $\mathbb R^N$?

as the title says. Let $A^n$ be an $n$-dimensional closed Alexandrov space. Does it admit a bi-Lipschitz embedding into the Euclidiean space $\mathbb R^N$ for sufficiently large $N$? I know there are ...
2
votes
1answer
194 views

Diameter estimate of distance sphere of positive curved manifold

Let $M$ be an $n$-dimensional Riemannian manifold with sectional curvature lower bound 1. Fix a point say $O\in M$, let $S(r)$ denote the distance sphere centered at $O$ with radius $r$. The classical ...
4
votes
1answer
152 views

Does convex set in Alexandrov space has positive reach?

Let $M$ be a metric space, $A$ a subset of $M$. The reach (defined by Federer) of $A$ in $M$ is the largest $r_0\ge 0$ such that if $x\in M$ and the $d(x, A)< r_0$, then $A$ contains a unique point ...
2
votes
1answer
163 views

Positivity of second fundamental form implies global convexity?

Let $M$ be a Riemannian manifold of dimension $n$. Let $N\subset M$ be a subset with smooth boundary $\Sigma=\partial N$. If one assume the second fundamental form $II$ with respect to inner normal ...
4
votes
2answers
276 views

Cutlocus and conjugate points

I am thinking about the following questions about the cutlocus of a point in a Riemannian manifold or of a hypersurface in the Euclidean space: 1) If all the points of the (nonvoid) cutlocus of a ...
6
votes
1answer
192 views

Fattening of totally convex sets

Suppose $(M, g)$ is an open complete nonnegatively curved Riemannian manifold with $d$ its distance. A totally convex set $C\subset M$ has the property that for any two point $x, y \in C$ any ...
1
vote
2answers
215 views

Examples on small cut radius of totally convex set in non-negatively curved manifold

Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...
4
votes
3answers
378 views

Degeneration of riemannian metrics with curvature bounds

In short, I'm curious to know what modes of degeneration of metric might still keep the curvature bounded. More precisely, assume we are keeping the total volume of the manifold fixed and deform the ...
4
votes
1answer
272 views

Collapsing of Riemannian manifolds with a group action

Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold ...
5
votes
1answer
264 views

Minimal distance spheres in complex projective spaces

My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold ...
5
votes
1answer
420 views

Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
9
votes
1answer
509 views

A strange question about closed geodesics on a closed manifold

I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me to know the answer to the following kinda weird question: Does there exist a closed Riemannian manifold $M$ ...
2
votes
1answer
378 views

Geometric conditions for isoperimetric, Sobolev, Poincar\'e inequalities on a riemannian manifold

By a theorem of Lichnerowicz, on a riemannian manifold $M^{(m)}$ with positive Ricci curvature, the reciprocal of Sobolev constant(ie. the first eigenvalue of laplacian) can be bounded from below by ...
3
votes
1answer
218 views

Constant Mean Curvature hypersurfaces “condensing” onto a minimal submanifold

Let $M$ be Riemannian manifold and $S\subset M$ a minimal submanifold, with $\dim S<\dim M-1$. According to a few references (e.g., Mahmoudi, Mazzeo & Pacard), it should not be hard to see ...
4
votes
0answers
595 views

“The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics”

The title is a quote from p.256 of Wilhelm Klingenberg's 1995 Riemannian Geometry (Google Books link): Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
9
votes
1answer
316 views

Length spectrum for Riemannian metrics in the projective plane

Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum? This question is related to MO questions Length spectrum and Zoll surfaces ...
3
votes
1answer
208 views

Length spectrum and Zoll surfaces of revolution

The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces, all of whose ...
9
votes
0answers
254 views

Is it overkill to invoke Kirszbraun theorem to prove the following fact ?

Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
3
votes
0answers
183 views

Seek “typical examples” for the structure of spaces with two-sided Ricci bounds

By a 1990 paper of Michael Anderson, the following is true: Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds ...
3
votes
0answers
167 views

Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds

I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...
4
votes
1answer
617 views

Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?

As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically, A complete connected Riemannian manifold ...
2
votes
1answer
251 views

The Tubular Neighborhood of a Closed Geodesic

Suppose $M_{g}$ is the mapping torus $\Sigma_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma_{g} \to \Sigma_{g}$ is an ...
16
votes
8answers
1k views

Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern ...
10
votes
3answers
1k views

Convex hull on a Riemannian manifold

Let $M$ be a complete Riemannian 2-manifold. Define a subset $C$ of $M$ to be convex if all shortest paths between any two points $x,y \in C$ are completely contained within $C$. For a finite set of ...
5
votes
0answers
467 views

Homometric $\Rightarrow$ isometric?

Suppose you know that there is a mapping between two Riemmanian manifolds $M_1$ and $M_2$ such that, for each $x_1 \in M_1$, the (codimension-1) measure of the set of points at distance $d$ from $x_1$ ...
1
vote
0answers
133 views

How to derive an energy measure of metric deforming

The problem is an abstract from applied science. Given an $n$ dimensional Riemann manifold with metric $\langle M, g\rangle$, we could define deformation of the metric $g(t)$ where $t\in [0,1]$, for ...
7
votes
1answer
646 views

Is there an elementary way to show the triangular inequality for this expression ?

Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
7
votes
2answers
265 views

Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension

In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the ...
5
votes
0answers
1k views

Geometric Proof that Fubini-Study Metric is Round

The Fubini-Study metric d(x,y) on $CP^1$ is defined as follows: for x and y in $CP^1$ let v and w be unit vectors in $C^2$ representing x and y. Then $d(x,y)=2arccos(\langle v,w\rangle)$. The round ...
9
votes
1answer
383 views

Are packing-homogeneous spaces homogeneous?

Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...
6
votes
3answers
868 views

What spaces have well known horofunctions?

Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the ...
38
votes
7answers
4k views

Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
6
votes
2answers
757 views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...