29
votes

Accepted

### Can we make distances in a finite subset of a manifold whatever we want?

It is not possible to find $5$ points $x_1,\ldots,x_5$ on a genus zero Riemannian 2-manifold (a sphere) such that $d(x_i,x_j)=1$ for all $i,j$.
The reason is that the complete graph $K_5$ is not ...

19
votes

Accepted

### When is a flow geodesic and how to construct the connection from it

Note: I've decided that this answer should be rearranged a bit
so that it clearly separates the discussion of the basic properties of
the tangent bundle from the discussion of the formulae ...

16
votes

Accepted

### Geodesic preserving diffeomorphisms of constant curvature spaces

For $\mathbb{R}^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $\mathbb{R}^n$ taking lines to lines are ...

11
votes

### When are geodesics straight lines?

The geodesics are straight lines, in geodesic normal coordinates, just when the associated projective connection is flat. See Kobayashi and Nagano, On projective connections, Journal of Mathematics ...

10
votes

Accepted

### Closed geodesics on constant positive Gauss curvature surfaces

The answers to your questions about geodesics on such surfaces can be found in A. L. Besse's book Manifolds all of whose geodesics are closed (1978, Springer Ergebnisse series). Especially, you ...

10
votes

Accepted

### Geodesic in space of circulant matrices

The answer is as follows: When $U$ is a positive-definite, symmetric, circulant matrix in $\mathrm{GL}(n,\mathbb{R})$, then there is a symmetric circulant matrix $u$ such that $U = e^u$ and the curve $...

10
votes

### Simple, closed geodesics in $\mathbb{S}^3$ manifold

Just to set the terminology, a closed geodesic is called simple when it is a smooth embedded circle in the Riemannian or Finsler manifold. If a closed Riemannian manifold is not simply connected one ...

9
votes

### Convexity in co-ordinate charts of geodesic balls

This is certainly true. If you choose $\epsilon>0$ smaller than half the injectivity radius of $g$ at $p$ (in particular, sufficiently small that the exponential map $\exp_p:T_pU\to U$ is well-...

8
votes

Accepted

### Simply connected manifolds with dense geodesics on the tangent bundle

Burns and Donnay proved that every surface (including a sphere) admits a Riemannian metric that makes the geodesic flow ergodic with respect to Liouville measure, and hence topologically transitive (...

7
votes

### When is a flow geodesic and how to construct the connection from it

Here is a geometric way that turns out to be equivalent to Robert's answer (i.e., to the Klein-Grifone-Foulon approach to connections associated to a second order ordinary differential equation of a ...

7
votes

Accepted

### Existence and uniqueness of geodesics in low regularity

There is a classical example by Hartman that shows failure of uniqueness for $C^{1,\alpha}$ metrics. (P. Hartman, On the local uniqueness of geodesics, Amer. J. Math. 1950).
You could lower the ...

7
votes

Accepted

### k-flats in homogeneous spaces

If the k-flats are compact, then the space must be symmetric
(Molina-Olmos, J. Differential geometry
45 (1997) 575-592; see also Proc. Amer. Math. Soc. 129 (2001), 3701-3709).
Homogeneous spaces (non-...

7
votes

Accepted

### Does every Zoll metric on $\mathbb{S}^2$ arise from a perturbation of the round metric?

The last I checked, it was unknown whether the set of Zoll metrics on the $2$-sphere was connected. What Guillemin (V. Guillemin, The Radon transform on Zoll surfaces, Advances in Mathematics 22 (...

7
votes

Accepted

### Geodesics for non differentiable riemannian metric

This is a length-metric, that is any two points $x$ and $y$ can be joined by a path with length arbitrary close to the distance from $x$ to $y$.
Further, your metric space is locally bi-Lipschitz to ...

7
votes

### Complete geodesics on hyperbolic a pair of pants

I suggest that you look at what happens in the universal cover given by the Poincare disk model. The five geodesics are quite easy to see. In the picture below, the greyed zone is a fundamental domain ...

7
votes

Accepted

### Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one ...

6
votes

Accepted

### The midpoint geodesic

The answer is no.
Define the manifold
Let $f$ be a smooth function on $\mathbb{R}$ satisfying
$f(r) = |r|$ for $|r| > 3$
$f(r) > 0$
$f''(r) \geq 0$.
The corresponding warped product metric ...

6
votes

Accepted

### 3-manifolds with all geodesics closed

There are integral homology spheres in the following Thurston geometries: $S^3$, $\mathrm{PSL}(2,\mathbb{R})$, and $H^3$. No manifold of the latter two types (when equipped with any metric) can have ...

6
votes

Accepted

### Compactness theorem for minimal surfaces

What's going on with the picture you drew is that because the $g_i$ have regions of negative curvature (in the "valleys" where the $M_i$ sit) that become arbitrarily close to the "hilltop" where $M$ ...

6
votes

Accepted

### Hopping geodesics

Yes, such examples do live. The following answer was given in "Metric spaces of non-positive curvature" by Bridson and Haefliger; thanks to GGT and Moishe Kohan.

6
votes

### Metrics on torus without closed contractible geodesics

This is a comment, not an answer, but it's too long to fit into a comment window.
I don't know of a 'generic' condition on metrics on the torus that would guarantee that there are no null-homotopic ...

6
votes

### Is there the longest geodesic?

No such bound $C(\gamma)$ exists, even when $S$ is the two-sphere and even assuming that all geodesics considered are simple. Here is the example (which generalises to surfaces with genus).
Suppose ...

6
votes

### Vanishing Gaussian curvature

Yes.
Consider segments of $u$-coordinate curves between two $v$-coordinate curves $\{u=u_1\}$ and $\{u=u_2\}$. These segments are a family of geodesics of the same length $|u_1-u_2|$. By the first ...

5
votes

Accepted

### Primary definition of a geodesic

I think that depending on what is the most fundamental structure you consider in your Riemannian manifold, both answers can be true. Let me explain.
In Euclidean geometry, one can consider the affine ...

5
votes

Accepted

### Does there exist a closed geodesic go through a $\epsilon$-net of a hyperbolic surface?

Yes, such a closed geodesic always exists. See Theorem 1.1 of this paper by Basmajian, Parlier, and Souto (which I found by searching under the term "density of closed geodesics on a hyperbolic ...

5
votes

### Primary definition of a geodesic

I have never seen the term "geodesic" explained in a hand-waving fashion in any other way than as a shortest path curve. From this (meager) evidence, I am willing to assume that, historically, the ...

5
votes

Accepted

### Thrice intersecting closed geodesic on genus 2 orientable closed surface

Yes, such curves exist on closed hyperbolic surfaces.
As mentioned by Sam Nead, one can think of such a curve as lying on a subsurface which is a 4-holed sphere or 2-holed torus (genus one with two ...

5
votes

### Bounding probability densities on a Wasserstein-2 geodesic

It turns out that for most convex domains $\Omega$, one can find smooth probability measures $\rho_0$ and $\rho_1$ which are supported on $\Omega$, have strictly positive density everywhere on $\Omega$...

5
votes

### Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

I did a bit of a literature search and couldn't find much.
Let $\Sigma$ be a closed surface, and $PT\Sigma$ the projective tangent bundle (double covered by the unit tangent bundle $X$). Lenzhen and
...

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