# Questions tagged [geodesics]

The geodesics tag has no usage guidance.

130
questions

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### 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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55 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 ...

**5**

votes

**1**answer

165 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 ...

**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.
...

**2**

votes

**1**answer

208 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 ...

**4**

votes

**1**answer

121 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(\...

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57 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 ...

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50 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**

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55 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,...

**1**

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**3**answers

245 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 ...

**3**

votes

**1**answer

177 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**

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103 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 ...

**9**

votes

**3**answers

603 views

### Is there the longest geodesic?

Given a closed 2-surface $M$ together with a Riemannian metric $g$.
We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$.
...

**2**

votes

**1**answer

96 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**

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94 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}...

**3**

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**1**answer

90 views

### Almost geodesic on non complete manifolds

Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\...

**3**

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162 views

### Infinitely many simple closed geodesics in any compact orientable surface but the sphere

My question is the following: if $(\Sigma, g)$ is any compact orientable Riemannian surface of genus $g \geq 1$, is it true that there are infinitely many closed, simple and geometrically distinct ...

**3**

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48 views

### Semiclassical analysis and reflection law

I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...

**2**

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**1**answer

103 views

### Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as:
A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...

**7**

votes

**1**answer

350 views

### Complete geodesics on hyperbolic a pair of pants

I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:
I am trying to understand the article by Maryam ...

**2**

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196 views

### Geodesics and potential function

I try to assemble concepts of differential geometry for my own comprehension of the subject. I understand a manifold is a higher dimensional surface. It has a metric which perform inner product in the ...

**3**

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**1**answer

74 views

### On a geodesic mapping of a square

Let $X$ be a proper geodesic space which is uniquely geodesic. Let $\phi:[0,1]\times[0,1] \to X$ be a function which satisfies the following:
The maps $\phi(0,\cdot)$, $\phi(\cdot,0)$, $\phi(1,\cdot)$,...

**3**

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77 views

### Almost-geodesics on a Riemannian Hilbert manifold which are still almost geodesics in some submanifold

Let $H$ be a separable infinite dimensional Hilbert space, and consider it as a Hilbert manifold in the usual way (that is, with the single chart with the identity map). It is known that there always ...

**13**

votes

**1**answer

400 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 ...

**0**

votes

**1**answer

140 views

### Riemannian metrics on matrix space for which the restriction of trace function to each complete geodesic is a bounded function

Edit: According to comment by Leo Monsaingeon I revise my question:
Is there a Riemannian metric on $M_n(\mathbb{R})$ for which the function $trace$ is a bounded function on every complete(whole)...

**7**

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**1**answer

319 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 ...

**3**

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101 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 ...

**5**

votes

**2**answers

241 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 ...

**2**

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38 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 ...

**3**

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**1**answer

387 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 \...

**17**

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**1**answer

320 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 [...

**12**

votes

**1**answer

812 views

### Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...

**1**

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64 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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**2**answers

301 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 ...

**5**

votes

**2**answers

558 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 \...

**7**

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93 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 ...

**18**

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**2**answers

925 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....

**3**

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38 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(\...

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235 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\...

**6**

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**1**answer

396 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 ...

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112 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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212 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 ...

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288 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 ...

**3**

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137 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 ...

**8**

votes

**1**answer

172 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 ...

**2**

votes

**0**answers

66 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**

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**1**answer

169 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 $X$? ...

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241 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 ...

**7**

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**1**answer

302 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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56 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 ...