Questions tagged [geodesics]

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3
votes
0answers
94 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 ...
7
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0answers
139 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 unstretchable, ...
2
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0answers
28 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 ...
5
votes
1answer
195 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
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1answer
297 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
1answer
132 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 \...
0
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0answers
45 views

Conjugate points and Jacobi matrices

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 3$ and let $\gamma:[-2,2]\to M$ be a geodesic that does not contain any conjugate points on $[-2,2]$. I have two questions, as ...
6
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0answers
73 views

Blaschke points

A Blaschke point of a metric space is a point so that every geodesic (i.e. locally shortest path) starting at that point and of length less than the diameter of the metric space is the unique shortest ...
2
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0answers
31 views

Conjugate points for a family of generalized curves

Let $(M,g)$ denote a compact smooth Riemannian manifold with boundary and let $\mathscr F$ denote a family of smooth curves $\gamma$ such that they solve $$ \nabla^g_{\dot \gamma} \dot\gamma = F(\...
1
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0answers
67 views

time-derivative and differential of a geodesic flow

I came across the following question in relation to another question. Let $X\colon M \to TM$ be a vector field over a manifold $M$ and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" for the vector ...
2
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0answers
183 views

A geometric property of certain Lie groups

I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive ...
0
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0answers
184 views

directional derivative along geodesic flow of vector field

A rather elementary question for the differential geometers. Let $M$ be a Riemannian manifold, let $X\colon M \to TM$ be a vector field, and let $$\phi_t = \exp(tX)$$ be the "geodesic flow" of the ...
8
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1answer
153 views

Geodesic preserving diffeomorphisms of constant curvature spaces

Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$. I would like to have a classification of all diffeomorphisms $X\to X$ which map ...
3
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0answers
121 views

Is there a family of Riemannian manifolds with explicitly solvable geodesics for manifold M?

By this paper https://www.mat.univie.ac.at/~michor/rie-met.pdf for any given manifold $M$, there is a Hilbert manifold (I believe its a Hilbert manifold, if not it is still an infinite dimensional ...
2
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0answers
41 views

Variational inference: Does the natural gradient follow (Fisher-Rao) geodesics locally?

Amari's natural gradient descent is a well-known optimisation algorithm for functionals defined on statistical manifolds. It consists of preconditioning the vanilla gradient descent update rule with ...
0
votes
1answer
165 views

What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure?

Consider $X = (\mathbb{R^n},c)$, where $c$ is the equivalence class of all torsion free affine connections having straight lines as unparameterized geodesics. What is the group of symmetries of $X$? ...
3
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0answers
132 views

Geodesics (Local vs Global)

Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
4
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2answers
269 views

Lipschitz constant of exponential map

I asked before this question on MSE but I was not able to work out the details on my own. Suppose $M$ is a smooth compact Riemannian manifold, take $p \in M$ and consider the map $$ T_pM \ni v \...
4
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0answers
233 views

Filling geodesic triangles by geodesic segments

Let $ M $ be a Hadamard manifold. For every $ x,y\in M $, define $[x,y]$ to be the geodesic segment connecting $ x $ to $ y $, as a subset of $ M $. Let $ x,y,z\in M $ be arbitrary and consider $p\...
5
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1answer
211 views

Compactness theorem for minimal surfaces

I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here. Theorem: Let $\Omega$ be an open subset of a Riemannian $3$-manifold. ...
3
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0answers
55 views

Transformation between nearby tangent planes [closed]

This question is kinda long, but the picture is quite clear. Question: Let $(M,g)$ be a Riemannian manifold, $p$ a point on $M$, $U$ an open neighborhood of $0\in T_pM$ such that $exp_p|_U$ is a ...
4
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0answers
105 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 ...
6
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1answer
300 views

Characterisation of Sobolev Spaces on manifolds of bounded geometry via geodesic coordinates

I have a reference request concerning equivalent norms on Sobolev Spaces on manifolds of bounded geometry. This may be obvious to the experts but I am not working in the field and only want to use ...
6
votes
0answers
154 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 $\...
6
votes
2answers
276 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 ...
3
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0answers
88 views

non-self-intersecting geodesics

Suppose $(M,g)$ is a smooth compact orientable Riemannian manifold of dimension $d \geq 3$ with a smooth boundary $\partial M$ and let $\gamma$ be a maximal geodesic in $M$ starting from a point $p \...
3
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0answers
83 views

What kind of set is this, spanned by two positive definite matrices?

Let $A$ and $B$ be Hermitian positive definite $n\times n$ matrices over $\mathbb C$ or $\mathbb R$. Then for real $k,\ell,$ the matrix $A^kB^\ell A^k$ is well-defined and again Hermitian positive ...
13
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4answers
548 views

Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...
20
votes
2answers
874 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. ...
5
votes
1answer
92 views

Decomposition of a Jacobi field along a lightlike geodesic

Consider a Lorentzian manifold of dimension $1+n$ (with $n\geq1$) and a lightlike geodesic $\gamma(t)$ on it. One can define a Jacobi field $J(t)$ along $\gamma$ in the usual way without issues. In ...
11
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2answers
441 views

Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice. I have heard that, for any real number $\...
5
votes
2answers
198 views

Bounding probability densities on a Wasserstein-2 geodesic

Consider two probability measures which are supported on a bounded domain $\Omega$ with density functions $p_0$ and $p_1$. It is well-known that for the Wasserstein-2 distance, there exists uniquely a ...
5
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2answers
182 views

Thrice intersecting closed geodesic on genus 2 orientable closed surface

Does there exist a closed geodesic on a closed genus 2 orientable surface (with hyperbolic metric) that self-intersects at only one point thrice?
5
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0answers
168 views

Higher order variations of Riemannian geodesics

Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic. There is a well established theory of the first order ...
1
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0answers
53 views

Conformal factors and light rays

Suppose $(\mathcal{M},g)$ is a $3$-dimensional Riemannian manifold and let $\gamma \in \mathcal{M}$ denote an arbitrary curve in $\mathcal{M}$. Does there exist a conformal factor $c>0$ such that $...
3
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0answers
73 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 ...
13
votes
1answer
431 views

Regularity of geodesics

If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample. For a related discussion see Metric ...
2
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0answers
168 views

Understanding the domain of the “volume density function” on Riemannian manifold

I have some trouble on understanding the domain of the "volume density function" on Riemannian manifold. Putting the volume density function in quote means actually I am working on the function ...
8
votes
1answer
322 views

Closed geodesics on constant positive Gauss curvature surfaces

Can we show that geodesics with a rational radius $ r_{mid-equator} =a q/p \,(p>q ) $ at mid-equator on a spindle type surface of revolution of constant Gauss curvature $ (K=1/a^2 \, $in $\, \...
2
votes
1answer
431 views

When are geodesics straight lines?

Suppose I have a global coordinate system on a manifold, which is affine with respect to an affine connection on that manifold. The connection is flat and torsion free, and the connection coefficients ...
11
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1answer
503 views

Length decreasing homotopies of curves

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement: Statement. ...
4
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1answer
296 views

3-manifolds with all geodesics closed

A theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \...
4
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0answers
351 views

Graph contained in a metric space

I have a metric space $X$ and a graph $G=(V,E)$ whose set of vertices is a subset $V\subset X$ (and $E$ is the set of edges, which is a symmetric subset of $V\times V$). For each $v\in V$, the set of ...
14
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0answers
466 views

How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?

There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $\mathbb{S}_n \to \mathbb{R}^{n+1}$. Is ...
2
votes
1answer
292 views

On triangle comparison in Riemannian manifolds with upper sectional curvature bound

I have a question on Riemannian geometry or CAT(k) geometry, which might be simple for experts. Suppose $M$ is a complete smooth Riemannian manifold with sectional curvature bounded from above by $k&...
2
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0answers
62 views

Continuous dependence on initial data for (null) geodesics in Lorentzian manifolds

I'm looking for a reference for the following theorem: Theorem Let $\mathcal M$ be a smooth manifold with smooth Lorentzian metric $g$. Let $p$ and $q$ be two points on the same (causal) geodesic, ...
4
votes
1answer
131 views

Can geodesics be acyclically directed?

Let $M$ be a length space, and let $\Pi$ be a finite set of geodesics in this space that are the unique geodesic between their endpoints. Additionally, suppose that the intersection of any two ...
10
votes
1answer
467 views

Existence and uniqueness of geodesics in low regularity

Consider a Riemannian manifold $(M,g)$. How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to ...
5
votes
1answer
212 views

Does there exist a closed geodesic go through a $\epsilon$-net of a hyperbolic surface?

An $\epsilon$-net of a closed hyperbolic surface $X$ is a finite set of points $p_i$ such that the family of balls centered at $p_i$ with radius $\epsilon$ is a cover of $X$, and the family of balls ...
4
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0answers
65 views

Do geodesics on a hyperkähler quotient have nice lifts?

Suppose one has a flat quaternionic vector space $V$, with a compatible inner product $g$. So $(V,g)$ is a flat hyperkähler manifold. Assume that there is some compact Lie group $G$ acting on $V$ ...