4
votes
0answers
223 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 $\sqrt{2dx^2 + 3dy^2}$ ...
7
votes
1answer
217 views

Smoothing of piecewise Euclidean Riemannian metrics

Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
4
votes
0answers
126 views

Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?

Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...
2
votes
2answers
535 views

Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry", an English version ...
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 $M$ ...
0
votes
1answer
167 views

relation with jacobifields in a small neighbourhood

hi, I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
4
votes
2answers
335 views

Conformal structure does not see conical singularities

the conformal structure does not see the conical singularities of a polyhedral surface. This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli). The sentiment is ...
3
votes
4answers
1k views

space of geodesics

hallo, i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | ...
5
votes
1answer
917 views

Good Surface,Bad Surface-Surface classification

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help. We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...
12
votes
3answers
628 views

F→E→B bundle with B,E,F hyperbolic: possible?

It would be interesting to me obtain an answer to the following easy to state question: Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...
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 $x_1$ ...
8
votes
4answers
2k views

Eigenvalues of Laplacian-Beltrami operator

I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value. I am interested in the case when ...
9
votes
2answers
817 views

Midpoint geodesic polygon / Birkhoff curve shortening

I would like to know under what conditions the process of creating a midpoint piecewise geodesic polygon converges on a surface $S \subset \mathbb{R}^3$. $S$ may be assumed smooth, closed, and ...
13
votes
3answers
514 views

Effective contraction of a loop. Reference or a simple proof?

Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$. Is there a base point preserving retraction of ...
8
votes
2answers
548 views

Illuminating piecewise-flat manifolds with geodesics

If one issues a geodesic in every direction from a point $p$ on a piecewise-flat 2-manifold, will it necessarily illuminate the entire surface? I know the answer is 'No,' but I would like to explore ...
16
votes
2answers
2k views

A question about the proof of Mostow rigidity

I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...
6
votes
4answers
1k views

Existence of Fermi coordinates on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there ...
9
votes
1answer
626 views
3
votes
1answer
401 views

The orthodrome of n-spheres.

I am a Computer Science undergraduate who does a lot of other tinkering in his free time. Right now, I'm tinkering with n-spheres. Specifically, I'm looking at the distances between a collection of ...
14
votes
4answers
2k views

Largest hyperbolic disk embeddable in Euclidean 3-space?

Hilbert proved that there's no complete regular (C^k for sufficiently large k) isometric embedding of the hyperbolic plane into R^3. On the other hand, the pseudosphere is locally isometric to the ...