Questions tagged [geodesics]

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56 views

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

Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...
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1 vote
0 answers
26 views

Regularity of the distant function in a conformal gauge

Suppose that $(M,g)$ is a smooth compact Riemannian manifold and let $c \in C^2(M)\cap W^{3,\infty}(M)$ be a positive function. Let $p\in M$ be fixed, let $U$ be a sufficiently small neighbourhood of $...
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5 votes
0 answers
147 views

Do geodesics avoid regions where the curvature diverges?

Let $(M^2,g)$ be a Riemannian manifold, with manifold boundary $\partial M$. We assume that the metric degenerates at the boundary, in the sense that the (Gauss) curvature diverges like $K \to +\infty$...
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1 vote
0 answers
58 views

Translate of a geodesic that goes through a fixed point on $\mathbb{H}$

Consider the complex upper half plane $\mathbb{H}$ with the hyperbolic geometry. Fix a point $z \in \mathbb{H}$ and also a geodesic $c$. I want to find a hyperbolic translation $\gamma c$ passes that ...
  • 549
6 votes
1 answer
100 views

On properties of Besse spheres

Let $(\mathbb{S}^2,g)$ be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and ...
1 vote
0 answers
57 views

Showing bound $\|\nabla_t \dot{\tilde{x}}_Y(h, 0)\| \le L \|Y\|_{2, \infty}$ for smooth homotopies of geodesics

This question pertains to Lemma 3.5 of this article. Let $M$ be a smooth Riemannian manifold and $\gamma$ some geodesic with respect to the Levi-Civita connection $\nabla$. For any $C^2$ vector field $...
0 votes
0 answers
94 views

Geodesics and gradient flow

Is there a construction in Riemannian geometry which relates the gradient flow of a function on a manifold with a certain metric with geodesics on another related manifold with its own metric?
1 vote
1 answer
98 views

A question on convexity and conjugate points

Let $(M,g)$ be a compact smooth simply connected Riemannian manifold with a smooth boundary. Assume also that $(M,g)$ does not have any conjugate pairs of points. Let $\Gamma \subset \partial M$ be a ...
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2 votes
0 answers
50 views

Image of tori in locally symmetric spaces and homology

Suppose we have a reductive group $G$ over $\mathbb{Q}$, a compact subgroup of the adelic points $K_f\subset G(\mathbb{A}_{\mathbb{Q}})$, and the associated locally symmetric space $$Y_K := G(\mathbb{...
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3 votes
0 answers
39 views

Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
1 vote
0 answers
69 views

What does the boundary of convex hulls look like in matrix Lie groups?

Let $G$ be a compact matrix Lie group under the Killing form metric $\langle \xi, \eta \rangle_g = -\frac{1}{2}\text{tr}((g^{-1}\xi)^T(g^{-1}\eta))$ for $g \in G$ and $\xi,\eta \in T_gG$. Let $C \...
3 votes
0 answers
71 views

Application of Santalo’s formula

Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that $$ \int_I f(\gamma(t))\,dt=0,$$ for any ...
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2 votes
0 answers
79 views

Smoothness of distance function induced by Finsler metric

Consider $\mathbb R^N$ endowed with a smooth Finsler metric $\phi:\mathbb R^N\times S^N\to (0,+\infty]$. The smoothness assumption are both on $\phi(x,\cdot)$ (being at least $C^{2,1}$) and $\phi(\...
22 votes
2 answers
1k views

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

Given a connected smooth manifold $M$ of dimension $m>1$, points $p_1,\dots,p_n\in M$ and positive values $\{d_{i,j};1\leq i<j\leq n\}$ satisfying the strict triangle inequalities $d_{i,j}<d_{...
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2 votes
0 answers
79 views

Distance and initial velocity of the shortest path along a smooth curve in a manifold

Let $(M,g)$ be a Riemannain manifold and let $p\in M$. Let $\gamma:[0,1] \to M$ be a smooth curve and let $p \notin \gamma([0,1])$. Assume further that for each $t \in [0,1]$ there is a unique (unit ...
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3 votes
0 answers
70 views

Geodesic foliations of open manifolds foliated by hyperbolic spaces

It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700). Suppose a complete ...
3 votes
1 answer
168 views

Infinite number of closed geodesics on distorted sphere

I would appreciate a reference to support this statement that appears under the Geodesic entry of the CRC Encyclopedia of Mathematics: "no matter how badly a sphere is distorted, there exists an ...
2 votes
0 answers
91 views

Comparison of sum of vectors and exponential map on a Riemannian manifold

Suppose $M$ is a simply-connected complete Riemannian manifold with bounded sectional curvature $\delta \leq K \leq \Delta < 0$. Let $p_0\in M$. Define the sequence of points $p_1, \ldots, p_n$ by $...
1 vote
0 answers
67 views

Completeness of infinitely intersecting causal geodesics in strongly causal spacetimes

Let $(M,g)$ be a connected, smooth, strongly causal Lorentzian manifold, and consider an inextendible causal geodesic $\sigma : [0,b) \to M$ (a priori, $b$ may be $\infty$) with the following property:...
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1 vote
0 answers
81 views

Injecitivity radius of Sasaki metric

Suppose we have a compact riemannian manifold $(M,g)$ and we endow $TM$ with the Sasaki metric $\tilde g$. Now I am interested in understanding the injectivity radius of $(TM,\tilde g)$ but I am ...
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2 votes
2 answers
257 views

Direct calculation of the Fisher distance via Riemannian geodesics

I'm looking for a reference for a direct calculation of the Fisher distance (to avoid overloading the term "metric") $d_F(x,y) := 2 \cos^{-1} \sum_i \sqrt{x_i y_i}$ as the geodesic distance ...
9 votes
0 answers
322 views

Volume of a geodesic ball in $\operatorname{SL}(n) / {\operatorname{SO}(n)}$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son Question: What is the volume of a ...
1 vote
0 answers
69 views

Dirichlet-to-Neumann map for second order ODE

Problem statement In a problem of interacting particles, I encountered a type of geodesic equation in $\mathbb{R}^n$ with an additional rotation and dilation term $$ \ddot\gamma(t) + e^{t Q} \Lambda ...
2 votes
1 answer
218 views

If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

This is a cross-post. Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric. Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
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4 votes
1 answer
138 views

What is the minimal length of a “Diagonal” in a Torus?

Given a Riemannian torus $(T,d)$ with fundamental group $\pi_1(T)=\langle a,b \mid ab=ba \rangle$. Denote for any $\gamma \in \pi_1(T)$ the infimum length of all representatives of $\gamma$ by $L(\...
1 vote
0 answers
96 views

Conditions under which a metric on a Riemannian manifold is induced by a Riemannian metric

Let $(M, g)$ be a Riemannian manifold. Lately, I've grown interested in what you may call a "modified geodesic" problem. Given some smooth, non-negative scalar field $V$ on $M$ (aptly called ...
1 vote
0 answers
67 views

Normal geodesic coordinates on submanifold comparison of coordinates

I would like to a find a formula which relates the normal geodesic coordinates associated to a submanifold to the geodesic coordinates on the manifold. More precisely, let $X$ be a closed submanifold ...
3 votes
0 answers
60 views

Semiconcavity estimate for the squared distance on a compact Riemannian manifold

I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...
3 votes
1 answer
240 views

Vanishing Gaussian curvature

I encounter the following claim in my general relativity research: Given a regular surface $x(u,v)$ such that in a neighborhood $V$ os some point $p$ in $S$, the coordinate curves $x(u,v=v_{0})$ and $...
5 votes
1 answer
178 views

Sufficient condition for geodesic convexity/connectedness

Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ...
4 votes
0 answers
150 views

Can the existence of geodesics be deduced from properties of the Laplacian?

As I understand it, the semiclassical trace formula in particular relates lengths of geodesics to eigenvalues of the Laplacian. Is it possible to prove that every compact Riemannian manifold has a ...
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1 vote
3 answers
312 views

Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths

While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any ...
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2 votes
1 answer
136 views

Hyperbolic length of curve that does not enter a collar

Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
1 vote
0 answers
110 views

Uniform divergence of geodesics in RAAGs

$\DeclareMathOperator\div{div}$Let $(X,d)$ be a metric space and let $\gamma$ be a geodesic in $X$. Roughly speaking, the divergence of $\gamma$ at a point $x\in \gamma$ is a function $\div:\mathbb{R}...
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9 votes
3 answers
668 views

Is there the longest geodesic?

Given a closed 2-surface $M$ together with a Riemannian metric $g$. We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$. ...
3 votes
1 answer
102 views

Almost geodesic on non complete manifolds

Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\...
4 votes
0 answers
189 views

Infinitely many simple closed geodesics in any compact orientable surface but the sphere

My question is the following: if $(\Sigma, g)$ is any compact orientable Riemannian surface of genus $g \geq 1$, is it true that there are infinitely many closed, simple and geometrically distinct ...
3 votes
0 answers
59 views

Semiclassical analysis and reflection law

I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
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7 votes
1 answer
489 views

Complete geodesics on hyperbolic a pair of pants

I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here: I am trying to understand the article by Maryam ...
2 votes
1 answer
120 views

Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
2 votes
0 answers
227 views

Geodesics and potential function

I try to assemble concepts of differential geometry for my own comprehension of the subject. I understand a manifold is a higher dimensional surface. It has a metric which perform inner product in the ...
3 votes
1 answer
91 views

On a geodesic mapping of a square

Let $X$ be a proper geodesic space which is uniquely geodesic. Let $\phi:[0,1]\times[0,1] \to X$ be a function which satisfies the following: The maps $\phi(0,\cdot)$, $\phi(\cdot,0)$, $\phi(1,\cdot)$,...
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4 votes
0 answers
108 views

Almost-geodesics on a Riemannian Hilbert manifold which are still almost geodesics in some submanifold

Let $H$ be a separable infinite dimensional Hilbert space, and consider it as a Hilbert manifold in the usual way (that is, with the single chart with the identity map). It is known that there always ...
0 votes
1 answer
167 views

Riemannian metrics on matrix space for which the restriction of trace function to each complete geodesic is a bounded function

Edit: According to comment by Leo Monsaingeon I revise my question: Is there a Riemannian metric on $M_n(\mathbb{R})$ for which the function $trace$ is a bounded function on every complete(whole)...
3 votes
0 answers
111 views

Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
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13 votes
1 answer
447 views

Which convex bodies can be captured in a knot?

Which convex bodies can be captured in a knot? This question is based on the discussion in "Is it possible to capture a sphere in a knot?". We assume that the knot is made from ...
2 votes
0 answers
53 views

Calculate the Jacobian of a particular diffeomorphism of parallelizable manifold onto itself

Let $M$ be $d$-dimensional parallelizable manifold. Let $e_k(x)$ for $k=1, \dots , d$ be the smooth vector fields forming orthonormal basis in tangent bundle of 𝑀, these vector fields exist because ...
8 votes
1 answer
403 views

Metrics on torus without closed contractible geodesics

It is easy to see that any closed geodesic on a flat 2-torus is noncontractible. Further the same holds true for a torus of revolution. Indeed either a closed geodesic is a meridian and therefore ...
17 votes
1 answer
345 views

Hopping geodesics

Is there a complete metric space $X$ with the following property? For any pair of points $p,q\in X$ there is unique minimizing geodesic $[pq]_X$ that connects $p$ to $q$, but the map $(p,q)\mapsto [...
3 votes
1 answer
602 views

Geodesic convexity and the Geometric Hessian

This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \...