Questions tagged [geodesics]

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Geodesic laminations on the 4-punctured sphere

Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
Nelson Schuback's user avatar
2 votes
0 answers
361 views

Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
hseldon39's user avatar
2 votes
0 answers
77 views

Does Kobayashi isometry map preserve complex geodesics?

Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
Begginer-researcher's user avatar
2 votes
1 answer
71 views

Jacobi fields in singular metric on quotient space

Consider the square $\Omega = (0,\pi) \times (0,\pi/2) \ni (r,\theta)$ endowed with the Riemannian metric \begin{equation} f^2 \big(\mathrm{d} r^2 + \sin^2(r) \, \mathrm{d} \theta^2 \big), \end{...
Leo Moos's user avatar
  • 4,912
0 votes
1 answer
142 views

Going from piecewise to genuine geodesic without decreasing number of intersections?

Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$. Suppose there are two geodesic segments $\gamma_i : [...
Leo Moos's user avatar
  • 4,912
2 votes
1 answer
95 views

Why is the set of singular points of starlike boundary $\Gamma$ closed?

I'm reading Geometric Inequalities of Yu. D. Burago and V. A. Zalgaller. I don't understand why $E_1$ is closed in the proof of the following lemma. Several definition. Suppose $ \Omega $ is a ...
HeroZhang001's user avatar
0 votes
0 answers
56 views

Calculating the principal curvature of the geodesic sphere of radius $r$ in the space of dimension $n$ and of constant sectional curvature $c$

I'm reading "Riemannian Geometry" by Manfredo P. do Carmo and I'm trying to calculate the principal curvature of the geodesic sphere of radius $r$ in the space of dimension $n$ and of ...
HeroZhang001's user avatar
0 votes
0 answers
49 views

Large deviations for geodesic flows of Lie groups

Let $G$ be a Lie group with finite Haar measure $\mu$. Choose $g\in G$ uniformly according to $\mu$. Let $\Phi^s$ denote the geodesic flow. Has the following been studied $$\lim_{t\to\infty}\mu\left(\...
user479223's user avatar
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12 votes
0 answers
416 views

Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
BernyPiffaro's user avatar
0 votes
1 answer
169 views

Compute distance between geodesics and perturbed geodesics on a Riemannian manifold via Jacobi field $\vert J \vert$

I would like to pose a question regarding the distance between a geodesic $\gamma(t)$ and a perturbed geodesic $\gamma_{\epsilon}(t)$ on a Riemannian manifold. Specifically, is the distance controlled ...
lumw's user avatar
  • 111
4 votes
1 answer
209 views

Geodesics on orthogonal matrix

Let $ O(n) $ be the manifold of orthornormal matrix, i.e. $$ O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}. $$ Then $ O(n) $ is a submanifold of $ \mathbb{R}^{n\times n} $. On $ O(n) $, there is a ...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
222 views

A Question about an article by Birman, Series

Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
Amirhossein's user avatar
3 votes
0 answers
190 views

Sweeping out the disk: what comes out?

In 2008, Larry Guth gave a new proof of a theorem of Gromov about the min-max widths of the unit $n$-ball. This states that the $p$-parameter width $\omega_p(k,n)$ (of sweepouts with $k$-dimensional ...
Leo Moos's user avatar
  • 4,912
3 votes
0 answers
198 views

Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
Ali Taghavi's user avatar
4 votes
1 answer
158 views

What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...
Eduardo Longa's user avatar
5 votes
0 answers
81 views

Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that: Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes. The integral of the Gaussian curvature in $K$ is ...
C M's user avatar
  • 381
-1 votes
2 answers
255 views

Are geodesics necessarily embedded?

I would like to ask a very basic/naive question. Given a Riemannian or pseudo-Riemannian manidold equipped with the Levi-Civita connection, is it known that all solutions of the geodesics equation are ...
Anton Sergeev's user avatar
5 votes
2 answers
461 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 ...
AmorFati's user avatar
  • 1,349
9 votes
1 answer
336 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 $K \to +\infty$...
Leo Moos's user avatar
  • 4,912
1 vote
0 answers
74 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 ...
Melanka's user avatar
  • 577
6 votes
1 answer
116 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 ...
Eduardo Longa's user avatar
1 vote
0 answers
59 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 $...
infinitylord's user avatar
0 votes
0 answers
214 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?
mathuser128's user avatar
1 vote
1 answer
145 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 ...
Ali's user avatar
  • 3,995
2 votes
0 answers
64 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{...
xir's user avatar
  • 1,964
4 votes
0 answers
58 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 ...
Spencer Kraisler's user avatar
2 votes
1 answer
134 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 \...
Spencer Kraisler's user avatar
3 votes
0 answers
92 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 ...
Ali's user avatar
  • 3,995
2 votes
0 answers
123 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(\...
DDG's user avatar
  • 21
23 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_{...
Saúl RM's user avatar
  • 7,916
2 votes
0 answers
95 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 ...
Mathav's user avatar
  • 61
4 votes
0 answers
97 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 ...
Claudio Gorodski's user avatar
3 votes
1 answer
250 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 ...
Joseph O'Rourke's user avatar
2 votes
0 answers
123 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 $...
M.R.Karimi's user avatar
1 vote
0 answers
70 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:...
jawheele's user avatar
  • 111
1 vote
0 answers
117 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 ...
Someone's user avatar
  • 761
3 votes
2 answers
322 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 ...
Steve Huntsman's user avatar
9 votes
0 answers
463 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 ...
ccriscitiello's user avatar
1 vote
0 answers
77 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 ...
André Schlichting's user avatar
2 votes
1 answer
227 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 ...
Asaf Shachar's user avatar
  • 6,591
4 votes
1 answer
157 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(\...
Sebastian's user avatar
1 vote
0 answers
132 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 ...
infinitylord's user avatar
1 vote
0 answers
101 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 ...
user197284's user avatar
3 votes
0 answers
77 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,...
grogTheFrog's user avatar
3 votes
1 answer
294 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 $...
pureorapplied's user avatar
5 votes
1 answer
202 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 ...
Ettore Minguzzi's user avatar
4 votes
0 answers
189 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 ...
Vivek Shende's user avatar
  • 8,653
1 vote
3 answers
428 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 ...
aitfel's user avatar
  • 119
2 votes
1 answer
186 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 ...
user158773's user avatar
1 vote
0 answers
123 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}...
M. Dus's user avatar
  • 1,900