Questions tagged [geodesics]
The geodesics tag has no usage guidance.
168
questions
4
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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 ...
3
votes
1
answer
87
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 ...
2
votes
0
answers
373
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_{\...
2
votes
0
answers
79
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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
73
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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{...
0
votes
1
answer
144
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 : [...
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 ...
0
votes
0
answers
57
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 ...
12
votes
0
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418
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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 ...
0
votes
1
answer
177
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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 ...
4
votes
1
answer
220
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 ...
2
votes
0
answers
226
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) ...
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 ...
3
votes
0
answers
200
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 ...
4
votes
1
answer
158
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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 ...
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 ...
-1
votes
2
answers
261
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
482
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 ...
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$...
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 ...
6
votes
1
answer
116
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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
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 $...
0
votes
0
answers
223
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
148
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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 ...
2
votes
0
answers
66
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{...
4
votes
0
answers
60
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 ...
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 \...
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 ...
2
votes
0
answers
125
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(\...
23
votes
2
answers
1k
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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_{...
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 ...
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 ...
3
votes
1
answer
256
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
125
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
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:...
1
vote
0
answers
117
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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 ...
3
votes
2
answers
323
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
467
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
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0
answers
78
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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
227
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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 ...
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(\...
1
vote
0
answers
134
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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
104
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
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,...
3
votes
1
answer
298
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
204
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
192
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 ...
1
vote
3
answers
441
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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 ...
2
votes
1
answer
188
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
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}...