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 ...
- 43.2k
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 ...
- 108k
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 ...
- 26.3k
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 ...
- 26.3k
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 ...
- 108k
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 $...
- 108k
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-...
- 108k
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 (...
- 8,903
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 ...
- 45k
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-...
- 86
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 ...
- 13.5k
7
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 ...
- 28.1k
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 ...
- 659
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 ...
- 18.7k
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 ...
- 4,787
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 $M$ ...
- 2,478
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.
- 45k
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 ...
- 108k
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 ...
- 28.1k
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 ...
- 32.4k
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 $\Omega$...
- 6,001
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 ...
- 68.8k
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 ...
- 15.4k
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 ...
- 4,312
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 ...
- 3,081
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
...
- 68.8k
5
votes
Accepted
If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?
Using the structure equations, it is not difficult to show that, if $f:(M,g)\to(N,h)$ is a diffeomorphism of (not necessarily complete) connected surfaces that is affine in the OP's sense, i.e., $\...
- 108k
5
votes
Lengths of closed geodesics and geodesic segments
If you restrict yourself to simple (non-self-intersecting) closed
geodesics on convex surfaces, then a complete characterization is known, as described in these two papers:
(1) Protasov, Vladimir Yu. ...
- 151k
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