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Questions tagged [geodesics]

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20
votes
2answers
774 views

A problem on Gauss--Bonnet formula

While teaching a course in differential geometry, I came up with the following problem, which I think is cool. Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature. ...
2
votes
0answers
27 views

Decomposition of a Jacobi field along a lightlike geodesic

Consider a Lorentzian manifold of dimension $1+n$ (with $n\geq1$) and a lightlike geodesic $\gamma(t)$ on it. One can define a Jacobi field $J(t)$ along $\gamma$ in the usual way without issues. In ...
11
votes
2answers
385 views

Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice. I have heard that, for any real number $\...
3
votes
1answer
76 views

Bounding probability densities on a Wasserstein-2 geodesic

Consider two probability measures which are supported on a bounded domain $\Omega$ with density functions $p_0$ and $p_1$. It is well-known that for the Wasserstein-2 distance, there exists uniquely a ...
6
votes
2answers
146 views

Thrice intersecting closed geodesic on genus 2 orientable closed surface

Does there exist a closed geodesic on a closed genus 2 orientable surface (with hyperbolic metric) that self-intersects at only one point thrice?
5
votes
0answers
127 views

Higher order variations of Riemannian geodesics

Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic. There is a well established theory of the first order ...
1
vote
0answers
50 views

Conformal factors and light rays

Suppose $(\mathcal{M},g)$ is a $3$-dimensional Riemannian manifold and let $\gamma \in \mathcal{M}$ denote an arbitrary curve in $\mathcal{M}$. Does there exist a conformal factor $c>0$ such that $...
3
votes
0answers
67 views

How to find geodesics in a Randers spaces?

Consider a Randers space $(M,F)$ that is the solution of the zermelo's navigation problem associated to a wind $W$ which is homothety; $\mathcal{L}_Wh=\sigma h$, $\delta$ constant, on a Riemannian ...
12
votes
1answer
367 views

Regularity of geodesics

If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample. For a related discussion see Metric ...
2
votes
0answers
67 views

Understanding the domain of the “volume density function” on Riemannian manifold

I have some trouble on understanding the domain of the "volume density function" on Riemannian manifold. Putting the volume density function in quote means actually I am working on the function ...
8
votes
1answer
260 views

Closed geodesics on constant positive Gauss curvature surfaces

Can we show that geodesics with a rational radius $ r_{mid-equator} =a q/p \,(p>q ) $ at mid-equator on a spindle type surface of revolution of constant Gauss curvature $ (K=1/a^2 \, $in $\, \...
2
votes
1answer
297 views

When are geodesics straight lines?

Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...
11
votes
1answer
491 views

Length decreasing homotopies of curves

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement: Statement. ...
4
votes
1answer
250 views

3-manifolds with all geodesics closed

A theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \...
3
votes
0answers
133 views

Graph contained in a metric space

I have a metric space $X$ and a graph $G=(V,E)$ whose set of vertices is a subset $V\subset X$ (and $E$ is the set of edges, which is a symmetric subset of $V\times V$). For each $v\in V$, the set of ...
14
votes
0answers
387 views

How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?

There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $\mathbb{S}_n \to \mathbb{R}^{n+1}$. Is ...
2
votes
1answer
229 views

On triangle comparison in Riemannian manifolds with upper sectional curvature bound

I have a question on Riemannian geometry or CAT(k) geometry, which might be simple for experts. Suppose $M$ is a complete smooth Riemannian manifold with sectional curvature bounded from above by $k&...
2
votes
0answers
52 views

Continuous dependence on initial data for (null) geodesics in Lorentzian manifolds

I'm looking for a reference for the following theorem: Theorem Let $\mathcal M$ be a smooth manifold with smooth Lorentzian metric $g$. Let $p$ and $q$ be two points on the same (causal) geodesic, ...
4
votes
1answer
128 views

Can geodesics be acyclically directed?

Let $M$ be a length space, and let $\Pi$ be a finite set of geodesics in this space that are the unique geodesic between their endpoints. Additionally, suppose that the intersection of any two ...
9
votes
1answer
322 views

Existence and uniqueness of geodesics in low regularity

Consider a Riemannian manifold $(M,g)$. How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to ...
5
votes
1answer
160 views

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

An $\epsilon$-net of a closed hyperbolic surface $X$ is a finite set of points $p_i$ such that the family of balls centered at $p_i$ with radius $\epsilon$ is a cover of $X$, and the family of balls ...
4
votes
0answers
62 views

Do geodesics on a hyperkähler quotient have nice lifts?

Suppose one has a flat quaternionic vector space $V$, with a compatible inner product $g$. So $(V,g)$ is a flat hyperkähler manifold. Assume that there is some compact Lie group $G$ acting on $V$ ...
3
votes
0answers
77 views

A criterion for a differential equation to be realized as an Euler-Lagrange equation on the infinite dimensional space

I study PDEs that arise in fluid dynamics in an infinite dimensional Riemannian geometric perspective. For example, Ebin-Marsden(1970) showed that the group of volume preserving diffeomorphisms has an ...
3
votes
0answers
41 views

Geodesics in norm balls

Recently, some problems that I work on require that I understand a bit of hyperbolic complex geometry. Assume that $B \subset \mathbb{C}^n$ is the unit ball of some norm $\|\cdot\|$ (not induced by an ...
2
votes
0answers
67 views

Almost flat manifold crossing

Let $(M,g)$ be an almost flat Riemannian manifold with boundary, i.e. we have $diam(M,g) \leq 1$ and curvature $K \ll 1$. Let's suppose $M$ is diffeomorphic to $N \times [0,1]$ for some smooth ...
5
votes
1answer
192 views

Geodesic in space of circulant matrices

I'm trying to find the geodesic that connects the identity with some circulant, symmetric matrix $U\in\mathrm{GL}(N,\mathbb{R})$, meaning we have \begin{align} U=\left(\begin{array}{ccc} u_1 & ...
10
votes
3answers
436 views

Primary definition of a geodesic

I am wondering if there is a sense in which one of these definitions for a geodesic on a smooth Riemannian manifold is primary to the other. A geodesic has acceleration zero, i.e., it is self-...
2
votes
1answer
64 views

Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$

I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...
2
votes
0answers
113 views

Geodesics on Sp(2,R)

Given the symplectic group $\mathrm{Sp}(2,\mathbb{R})$, represented as real $2\times 2$ matrices, I would like to compute the geodesic from the identity matrix $1\!\!1$ to the group element \begin{...
1
vote
1answer
86 views

Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<...
0
votes
1answer
126 views

Rotation matrix between two column spaces

I would like to find a rotation matrix between two flats $F_1,F_2$ that are defined by the column spaces of the matrices $M_1,M_2 \in \mathbb{R}^{n \times k}$ ($k<n$) respectively. If it was to ...
5
votes
0answers
268 views

Examples of spiraling geodesics?

Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$ that has a geodesic $\gamma$ that spirals around a point $x$, getting closer and closer, but never reaching $x$? Here I ...
12
votes
2answers
630 views

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

Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following: If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, ...
1
vote
1answer
181 views

Squared distance function function from a submanifold

Let $S$ be a smooth closed submanifold of $M$. Let $U$ be a tubular neighborhood such that for any $x\in U\setminus S$ there is a unique minimizing geodesics. We now consider the distance squared to $...
-1
votes
2answers
317 views

Solving the geodesic equation for a singularity crossing curve

[Apologies if this question is not considered research level, but it received no substantive comments and no answers at math.SE; I thought it was straightforward, but maybe it isn't.] Part I - What ...
2
votes
1answer
103 views

Curvature and intersection of submanifolds

Let $(M,g)$ be a Riemannian manifold of dimension $n$. (In the case I am interested in, $M$ is a complex symmetric domain, but I do not think that this is relevant for the question.) Let $N$ be a ...
4
votes
1answer
111 views

Geodesic-like curves stemming from the heat kernel on a manifold

Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>...
4
votes
1answer
163 views

Simply connected manifolds with dense geodesics on the tangent bundle

A unit speed geodesic $\gamma:I\to M$ on a Riemannian manifold $M$ can be lifted to a curve $\sigma=(\gamma,\gamma')$ on $SM$, the unit sphere bundle (the unit tangent bundle) of $M$. Let us say that ...
6
votes
1answer
211 views

Geodesics for non differentiable riemannian metric

Let $M$ be a differentiable manifold of dimension $n>2$ with a Riemannian metric $g=\sum_{i,j=1}^ng_{ij}dx_idx_j$ such that in some points on $M$ its coefficients $g_{ij}$ are not differentiable (...
4
votes
1answer
236 views

The midpoint geodesic

Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment. Pick three distinct ...
7
votes
1answer
160 views

k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$. Question. Are ...
3
votes
1answer
117 views

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

The introduction here states 'A formal perturbation argument of Funk later indicated that, modulo isometries and rescalings, the general Zoll metric on $\mathbb{S}^2$ depends on one odd function $f:\...
2
votes
1answer
170 views

Convergence of Discrete Geodesic

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence: \...
1
vote
0answers
61 views

Existence of a brake geodesic on a non compact Riemannian mfd

I am interested how to find a geodesic (if it exists) on a Riemann manifold s.t. the geodesic connects 2 different points on the edge of the manifold the metric is positive definite everywhere on the ...
1
vote
1answer
82 views

Asymptotic formula for average geodesic length on graph?

If $G$ is a finite, connected simple graph then is there an expression for the average geodesic length? That is suppose I know two nodes $n_1$ and $n_2$, the number of edges in my graph and at those ...
4
votes
1answer
343 views

Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$. Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...
2
votes
0answers
124 views

Equivalence of local and global geodesics in projective spaces

Let $d:\mathbb R^n\times \mathbb R^n\to\mathbb R^n$ be a distance which induces the Euclidean topology and with which $\mathbb R^n$ is a length space. A continuous curve $\gamma:[a,b]\to\mathbb R^n$ ...
6
votes
0answers
194 views

Shortest path to inspect a polyhedron

This is a variant of two as-yet unsolved MO questions cited below. Let $P$ be a closed polyhedron in $\mathbb{R}^3$. The task is to find a shortest path $\sigma$ on the surface of $P$ from which all ...
0
votes
0answers
100 views

Proper time and asymptotic flatness

I have asked this question at physics stackexchange but got no response. I thought I could try my luck here: I'm trying to understand the concept of asymptotic flatness in general relativity, and ...
10
votes
0answers
476 views

Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...