Questions tagged [geodesics]
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185 questions
3 votes
1 answer
236 views
Are compact simple manifolds non-trapping?
A Riemannian manifold $(M,g)$ with boundary $\partial M$ is called simple if $\partial M$ is strictly convex (strictly positive second fundamental form), and for every two points $x,y\in M$ there is a ...
- 235
3 votes
0 answers
78 views
Two-point boundary problem for Jacobi fields on the Grassmann manifold
Jacobi fields on a Riemannian manifold can be expressed using the differential of the exponential map, given an initial value of the field $J(0)$ and its derivative $D_t J(0)$. Is it also possible to ...
- 159
3 votes
1 answer
195 views
Hyperbolic pants: boundary of collar neighborhood and shortest figure 8s
I'm looking for a construction of a $\mathbb{Z}/3$ symmetric pair of hyperbolic pants - all cuffs of length $a$ small, such that the shortest figure-8 geodesics in the surface have length bounded ...
- 453
2 votes
1 answer
219 views
Reconstruct the metric from partial information of the geodesic distance
Suppose in a Riemannian manifold $M$ we know the geodesic distance function $l(p_1,p_2)$ between any two points $p_1$ and $p_2$ (at least locally), we can reconstruct the metric as follows. We ...
- 121
0 votes
0 answers
149 views
Reflection on geodesic via group action
I'm trying to figure out, how to "mimic" the flow of a geodesic on the Stiefel Manifold via rotations. For this let $$ \operatorname{St}(n,k) = \{X \in R^{n \times k} | X^\top X = I_k\} $$ ...
- 1
22 votes
1 answer
2k views
Can a smooth function hide a point from the origin?
Suppose $f(x,y)$ is a smooth function sitting over the $xy$-plane. Assume $f(x,y)$ exists for all $x,y$. For example, $f(x,y) \,=\,\sin(x) \cdot \cos(y)$ as illustrated. For simplicity, assume $f(0,0)=...
- 153k
0 votes
1 answer
118 views
Totally geodesic connected submanifolds $M_1,M_2$ with $T_xM_1 = T_x M_2$ are equal?
Let $M$ be a smooth manifold with some torsion-free linear connection. Let $M_1,M_2 \subseteq M$ be totally geodesic, connected injectively immersed submanifolds which are complete (with respect to ...
- 684
1 vote
0 answers
180 views
Is there a variation of the Picard-Lindelöf theorem where the initial point is on the boundary of the open set $U?$ Details follow
Motivated by this question, and my own related research interest, I'm thinking to see under what conditions/hypotheses, the initial point in the Cauchy-Picard-Lindelöf theorem lies on the boundary ...
- 1,661
2 votes
0 answers
136 views
Space of singular geodesics in symmetric spaces of higher rank
It is known that for a large class of smooth manifolds $M$ (e.g. Euclidean space, hyperbolic space, the sphere, real, complex, quaternionic spaces, the Cayley plane, complex hyperbolic space) the ...
- 435
1 vote
0 answers
50 views
Cauchy problem for geodesics on a weakly convex Finsler manifold
By a Finsler manifold I shall mean a manifold $M$ equipped with an asymmetric norm $\|\cdot\|$ on each tangent space, such that $\|\cdot\|$ is smooth on the complement of the zero section. In ...
- 1,270
1 vote
0 answers
74 views
Obtaining the geodesic extension property by embedding in a larger space
Suppose $(X,d)$ is a Hadamard space. By considering basic examples like a compact interval in $\mathbb{R}$ or a closed unit ball in Hilbert space, $X$ need not have the geodesic extension property (...
- 163
2 votes
0 answers
125 views
Property of parallel translation in Green and Wu, "On the subharmonicity and plurisubharmonicity of geodesically convex functions"
In the mentionned paper, I am having difficulties in understanding the proof of lemma 2. Roughly, this lemma says that given any separation $\eta$ for the $C^\infty$ topology of smooth paths from $[-1,...
- 21
3 votes
2 answers
469 views
Lengths of closed geodesics and geodesic segments
Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties: existence of closed geodesics of arbitrarily long length on $M$...
- 31
7 votes
0 answers
196 views
Lonely globe trotters
In analogy with the lonely runners conjecture, imagine "globe trotters" each traveling on a longitudinal great circle at different (constant, positive) speeds. Each "trotter" ...
- 153k
1 vote
1 answer
222 views
An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$. We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
- 557
2 votes
0 answers
80 views
Connection vs Exponential preserving maps
Connection Preserving Diffeomorphisms The setting is a manifold $M$ equipped with a linear connection $\nabla$. Kobayashi & Nomizu [K&N §VI.1] define a connection preserving diffeomorphism (...
- 243
1 vote
1 answer
409 views
Derivatives and ODEs on Lie groups
I'll keep the question specific to the scenario I'm working with, which is the Lie group SO(3) and its Lie algebra so(3). Consider two rotation matrices $R_0$ and $R_1$, and define $R_t = \exp_{R_1} ((...
- 123
0 votes
0 answers
178 views
About geodesic vector fields and the status of a classic problem on the number of closed geodesics
A classical problem in differential geometry is to determine whether every compact Riemannian manifold admits infinitely many geometrically distinct closed geodesics. A good reference on the subject ...
- 1,853
4 votes
0 answers
186 views
Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics
Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies? Since $\mathrm{exp}_p$ is defined by the ...
- 339
4 votes
1 answer
213 views
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 ...
- 143
2 votes
0 answers
453 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_{\...
- 21
2 votes
0 answers
116 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\...
2 votes
1 answer
121 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{...
- 5,163
0 votes
1 answer
222 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 : [...
- 5,163
2 votes
1 answer
147 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 ...
- 197
12 votes
0 answers
473 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 ...
- 149
0 votes
1 answer
350 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 ...
- 111
4 votes
1 answer
343 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 ...
- 1,229
2 votes
0 answers
339 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) ...
- 231
3 votes
0 answers
230 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 ...
- 5,163
3 votes
0 answers
246 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 ...
- 290
4 votes
1 answer
197 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 ...
- 2,127
5 votes
0 answers
106 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 ...
- 391
-1 votes
2 answers
374 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 ...
5 votes
2 answers
1k 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 ...
- 1,399
9 votes
1 answer
385 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$...
- 5,163
1 vote
0 answers
106 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 ...
- 587
6 votes
1 answer
166 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 ...
- 2,127
1 vote
0 answers
80 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 $...
- 83
0 votes
0 answers
334 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?
- 41
1 vote
1 answer
197 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 ...
- 4,189
2 votes
0 answers
89 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{...
- 2,241
4 votes
0 answers
95 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 ...
- 528
2 votes
1 answer
204 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 \...
- 528
3 votes
0 answers
151 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 ...
- 4,189
2 votes
0 answers
187 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(\...
- 21
24 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_{...
- 13k
2 votes
0 answers
129 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 ...
- 61
4 votes
0 answers
130 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 ...
- 4,869
3 votes
1 answer
401 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 ...
- 153k