Questions tagged [geodesics]
The geodesics tag has no usage guidance.
169
questions
1
vote
1
answer
157
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} ((...
- 11.4k
0
votes
0
answers
58
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 ...
- 1,735
4
votes
0
answers
91
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 ...
- 2,758
2
votes
0
answers
379
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
3
votes
1
answer
93
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 ...
- 26.3k
2
votes
0
answers
82
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\...
- 60.2k
2
votes
1
answer
75
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{...
- 37.6k
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 ...
- 4,968
0
votes
1
answer
146
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 : [...
- 787
12
votes
0
answers
421
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
187
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
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$...
CommunityBot
- 1
4
votes
1
answer
232
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 ...
- 366
2
votes
0
answers
231
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) ...
- 103
3
votes
0
answers
201
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 ...
- 312
3
votes
0
answers
191
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 ...
- 4,968
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 ...
- 29.5k
2
votes
1
answer
135
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 \...
- 43.7k
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 ...
- 381
-1
votes
2
answers
265
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 ...
- 60.2k
5
votes
1
answer
165
views
Plane projection of Geodesics (Inverse view)
Maybe this question is so clear or maybe it is not exact. It is because of my very little knowledge of differential geometry. I am reading some material in this field and I got a question which seems ...
- 43.7k
2
votes
1
answer
303
views
2 questions about loops and negative curvature
$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$
1) will the geodesic in the free homotopy ...
- 43.7k
5
votes
2
answers
510
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 ...
CommunityBot
- 1
8
votes
1
answer
309
views
trapped geodesics
Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary. We call a point $p \in M$ regular if there exists a geodesic of finite arc length passing through $p$ with end points on $\...
- 43.7k
7
votes
2
answers
404
views
Convexity in co-ordinate charts of geodesic balls
Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$.
I would guess the following is true:
for $\epsilon$ sufficiently small, the $g$-geodesic ...
- 27.3k
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 ...
- 577
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 $...
- 63
1
vote
1
answer
149
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 ...
- 4,968
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 ...
- 4,968
0
votes
0
answers
227
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?
- 31
2
votes
0
answers
67
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{...
- 1,964
4
votes
0
answers
61
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 ...
- 518
3
votes
0
answers
94
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 ...
- 4,089
2
votes
0
answers
127
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
3
votes
0
answers
107
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 ...
- 60.2k
18
votes
2
answers
1k
views
Simple, closed geodesics in $\mathbb{S}^3$ manifold
Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p....
- 4,618
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_{...
- 42.5k
2
votes
0
answers
96
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
99
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 ...
- 4,674
3
votes
1
answer
264
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 ...
- 149k
2
votes
0
answers
129
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
$...
- 60.2k
1
vote
0
answers
71
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:...
- 111
3
votes
2
answers
330
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 ...
- 15.3k
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 ...
- 781
9
votes
0
answers
476
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 ...
- 11.4k
1
vote
0
answers
78
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 ...
- 1,081
5
votes
1
answer
205
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 ...
- 1,341
24
votes
3
answers
1k
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.
...
- 43.7k
2
votes
1
answer
228
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 ...
- 106k
4
votes
1
answer
160
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(\...
- 88.9k