# Questions tagged [geodesic]

A geodesic is a generalisation of the notion of a straight line to curved spaces, and can be sometimes be thought of as the locally shortest or extremal path between points.

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### 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 ...
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### Inversive product of tow ultraparallel geodesics in the hyperbolic plane is $\cosh{\rho}$

This is Lemma 7.17.3 in Beardon: Lemma 7.17.3 Let $L$ and $L'$ be geodesics in the hyperbolic plane. Then the inversive product $(L,L')$ is $\cosh{\rho(L,L')},1,\cos{\phi}$ according as $L,L'$ are ...
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### On the definition of pregeodesics as stationary points of the length functional in (pseudo-)Riemannian geometry

On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways: curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is ...
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### Does a coercive Riemannian metric imply geodesic completeness?

Let $M \subset \mathbb{R}^n$ be a bounded open (in the euclidean sense) set. Define a smooth function $Q:M \to S_{++}^n$, where $S_{++}^n$ denotes symmetric positive-definite $n \times n$ matrices. ...
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### Is parallel transport or a connection needed for geodesic computation?

I have been reading a bit more about differential geometry and I'm interested in it from the practical computational perspective. I have seen some places where it is mentioned that you cannot move ...
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### Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$

Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$ Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
1 vote
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### Geodesics connecting two points in the hyperboloid model of the hyperbolic plane

Let $\mathbb{H}^2$ be the upper sheet of the hyperboloid defined by $x^2+y^2-z^2=-1$ in three-dimensional Minkowski space $(\mathbb{R}^3, g_M)$, where $g_M = \text{diag}(1,1,-1)$. In other words, ...
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### Special orthogonal matrices, Geodesic and Manifold

My task is to find some manifold on stoichiometric matrices and find the geodesic distance between these nodes (not the euclidean distance). Here's my idea so far: Suppose we are given a matrix N of ...
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### Proof that geodesics have constant speed [duplicate]

Let $(M,g)$ be a Riemannian manifold. Let $(\varphi,U)$ be a chart. A curve on that chart $\gamma(t)=\varphi^{-1}(x^1(t),...,x^n(t))$ is a geodesic if it solves the geodesic equation \ddot{x}^i+\...
1 vote
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### Quasi-geodesic rays are closed to geodesic rays in proper hyperbolic geodesic spaces

We define the boundary of a hyperbolic metric space $\partial X$ as the equivalence classes of geodesic rays up to finite Hausdorff distance and $\partial_q X$ as the equivalence classes of quasi-...
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### Question on a stable geodesic lamination on a closed hyperbolic surface

Let me first state a theorem in Casson-Bleiler Automorphisms of Surfaces after Nielsen and Thurston. Theorem 5.5: Let $h:F\to F$ be a non-periodic irreducible automorphism of a closed orientable ...
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### Proof of length of a tangent vector of a geodesic is constant on Finsler Manifold

In proof of Proposition 2.2 of the book Introduction to Modern Finsler Geometry by Shen and Shen I have faced the following problem: Proposition 2.2: If $\sigma(t)$ is a geodesic on a Finsler manifold ...
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### Existence of geodesically convex neighborhoods in semi-Riemannian manifolds.

I am studying the text by Barrett O’Neill referenced below. On page 130, O'Neill states, as Proposition 7, that every point in a semi-Riemannian Manifold has a convex neighborhood. Convex is defined ...
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### Reparametrization of geodesics (especially in the context of geodesic flow)

The following is in the context of Riemannian geometry, but the pseudo-Riemannian case is just as important to me (not directly addressed here). I want to understand what freedom we have to ...
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### Can Jacobi Fields be used to explicitly calculate Lyapunov exponents for geodesic flows?

My background is in geometry, and I have just become interested in the ergodic theory of geodesic flow. In Sarnak's 1981 paper "Entropy estimates for geodesic flows," the following ...
1 vote
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### Ray Tracing In Mathematical Spaces

I really enjoyed the Not Knot video, but I don't fully understand the mathematics which is going on there. They are animating the space of the complement of the Borromean rings, but you can't just ...
1 vote
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### Marked points on fibers of $TS^2$ can trace out a helix during (parallel) transport.

It is claimed here that a mark on a rod parallelly transported by an observer moving along a geodesic $\gamma(t)$ on a smooth manifold $M$ (with a metric $g$) will trace out a helix, instead of a ...
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### Prove that if the cone is cut along a generator and flattened into a plane, then the geodesic becomes a straight line.

The is a problem from Differential Equations with Applications and Historical Notes by George F. Simmons. The chapter is about the calculus of variation. The hint is to represent equation of a right ...
1 vote
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### What type of curve follows a surface such that the line is straight when the surface is flattened

Given an arbitrary surface, a point on that surface, and a direction, what is the name of the type of curve which lies in the surface and which twists and turns along with the curvature of the surface,...
1 vote
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### Preconditioned gradient flow keeps boundedness

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a real analytic function with $\inf_{x\in\mathbb{R}^n} f(x) > -\infty$. If we know the solution $x:[0,\infty)\rightarrow\mathbb{R}^n$ to the vanilla ...
1 vote
I have a curve between two points $P_1 (x_1, y_1,z_1)$ and $P_2 (x_2, y_2, z_2)$ which both lie either on the surface of a cylinder (case $A$) or on the surface of a cone (case $B$). The curve is the ...