Questions tagged [geodesics]

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0answers
237 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 ...
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0answers
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 ...
2
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1answer
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
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1answer
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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0answers
51 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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0answers
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,...
3
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1answer
178 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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1answer
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 ...
5
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0answers
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 ...
1
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3answers
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 ...
2
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1answer
98 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 ...
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0answers
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}...
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3answers
606 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 $\gamma$. ...
3
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1answer
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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0answers
163 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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0answers
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 ...
7
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1answer
353 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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1answer
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 ...
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0answers
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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1answer
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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0answers
78 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 ...
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1answer
141 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)...
3
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0answers
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 ...
13
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1answer
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 ...
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0answers
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 ...
7
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1answer
321 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
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 [...
3
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1answer
394 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 \...
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0answers
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 ...
7
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0answers
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 ...
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0answers
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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0answers
114 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
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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0answers
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 ...
8
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1answer
173 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
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 ...
2
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0answers
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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1answer
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 $X$? ...
3
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0answers
243 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 ...
5
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2answers
567 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 \...
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0answers
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\...
7
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1answer
303 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
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 ...
4
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0answers
111 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
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 ...
6
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0answers
184 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
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2answers
302 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
85 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 ...