# Questions tagged [geodesics]

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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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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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### 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 ...
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### 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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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 $p\... 1answer 303 views ### Compactness theorem for minimal surfaces I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here. Theorem: Let$\Omega$be an open subset of a Riemannian$3$-manifold. ... 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 ... 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 ... 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 ... 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$\...
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 ...