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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Geodesic in local coordinates

A Riemannian manifold $\mathcal{X}$ contains, for certain coordinate chart $(U, \varphi)$, the (local) coordinate map $\varphi$ that maps a neighborhood of coordinate $p$ to an Euclidean space $\tilde{...
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Why are the red parts of this expression present $\int_a^b \sqrt{g_{cd}\frac{dx^c}{\color{red}{d\lambda}}\frac{dx^d}{\color{red}{d\lambda}}}d\lambda$?

I am now trying to teach myself about geodesics, and a passage of my notes reads: In this chapter geodesics have been introduced as generalizations of straight lines through $$\frac{Du^a}{ds}=0$$ ...
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Connection between the Euclidean metric $\overline{g}=\sum_{i}dx^idx^i$ and $\left<v, w\right>_\overline{g}=\sum_{i}v^iw^i, v, w\in T_x\mathbb{R}^n$

This is a quite elementary question, but I am aware that there are two equivalent ways of characterizing the Euclidean metric in a Riemannian manifold: 1.) $\overline{g}=\sum_{i}dx^idx^i$ and 2.) $\...
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Help with the proof that geodesics are locally minimizing

I am currently trying to understand the proof of the following statement from the Riemannian Geometry book from Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine. The statement is: Let $m_0 \in M$...
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How to show that geodesic in polar coordinates satisfies the great circle equation in cartesian

I've shown using Euler-Lagrange equation that the shortest path between two points on a unit sphere is $$\phi = -\arcsin\left(\frac{\cot \theta}{\sqrt{\frac{1}{c_1^2}-1}}\right) + c_2$$ Now, I need to ...
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Geodesic lines in Poincare disc model [duplicate]

There is Poincare disc model with metric $g=4\dfrac{dx^2+dy^2}{(4-x^2-y^2)^2}$ I have to find geodesic lines in this model. I tried with system of differential equation, but it is very complicated. I ...
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Solving geodesic equation numerically. How to deal with the arc length constraint?

The geodesic equation can be solved via some ode-solver, for example RK45. In each step I get the tangent vector as well as the new position. As the equation is only valid for arc length ...
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Is the convexity radius continuous?

Suppose $(M, g)$ is a Riemannian manifold. For each $x \in M$ define the $\textbf{convexity radius of $M$ at $x$}$, denoted by $\text{conv}(x)$ to be the supremum of all $\epsilon > 0$ s.t. there ...
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Is the map $T_X |_S (p) := \exp{X(p)}$ a diffeomorphism onto its image?

Preliminaries The exponential map $\exp : TM \rightarrow M$ is defined by $\exp{(v)} = \gamma_v (1) $ where $\gamma_v$ denotes the geodesic starting at $p \in M$ and initial velocity $v \in T_p M$. ...
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Continuity of the curve shortening process

I'm studying the shortening process, introduced in the book Course in Minimal Surfaces by T. Colding and W. Minicozzi, which is inspired by the Birkhoff's curve shortening process. In the book, is ...
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Geodesics on surfaces of centers of a surface

Ex. 16 of Ch. 9 from the book "Differential Geometry", by A. V. Pogorelov, goes as follows: "The surfaces $F_1$ and $F_2$ are called the surfaces of centers of the surface $F$ if they ...
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Conjugate points and expansion of the geodesic congruence

I am working in a Lorentzian manifold $(M, g)$ (but I think the problem would be quite similar in a Riemannian manifold) and I am considering a timelike geodesic whose tangent vector field is denoted ...
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Free boundary geodesics as a critical point of the energy functional

As a consequence of the formula for the first variation of the energy of a curve, we have the following known characterization of geodesics. A piecewise differentiable curve $c:[0,1]\to M$ is a ...
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Simple formulas for geodesics of the Beltrami-Klein model

Exercice : Show that the geodesics of the Beltrami-Klein model are the euclidean lines ? Let $\gamma_{p,T}(t)=\cosh(t)p+\sinh(t)T$ be a geodesic of the hyperbolic plane such that $\gamma_{p,T}(0)=p=(...
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Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
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Calculating the area of the hyperbolic triangle with vertices at $e^{i\pi/3}$, $i\sqrt2$, $i$ in the Upper Half Plane model

Sketch the geodesic triangle in H with vertices at $e^{i\pi/3}$, $i\sqrt2$, and $i$, and calculate its area. I'm not sure how to go about this question, any help would be great, thanks. I know I need ...
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A coordinate free computation of the acceleration of $ \exp_p(t(v + \frac{t}{2}w))$

$\newcommand{\al}{\alpha}$ Let $M$ be a smooth Riemannian manifold. Fix $p \in M, v,w \in T_pM$, and define $\gamma(t) = \exp_p(t(v + \frac{t}{2}w))$. Is there a coordinate-free proof that $(\nabla_{\...
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Let $S$ be a connected surface where the curvature of all its geodesic curves is constant, $K=1$. Prove that $S$ is contained in a sphere

Let we have $S$, a connected surface. Let's suppose that the curvature of all its geodesic curves is constant, $K=1$. I have to prove that $S \subseteq \mathbb{S}^2$. $\mathbb{S}^2=\{(x,y,z)\in \...
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Prove that if all geodesics of a connected surface $S$ has curvature 1 $S$ is contained in a sphere

I know that every geodesic curve of a connected surface $S$ has curvature 1. I want to prove that $S$ is contained in a sphere. I think that it can be useful to see that every point is umbilical to ...
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A fly sits on the outside surface of a cylindrical drinking glass . It must crawl to another point situated on inside of the glass . Find the shortest

A fly sits on the outside surface of a cylindrical drinking glass . It must crawl to another point situated on inside of the glass . Find the shortest path possible(neglecting the thickness of the ...
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Geodesics in Beltrami Poincare half plane model of the pseudo sphere?

Suppose points $\hat{a}$ and $\hat{b}$ on the pseudosphere correspond to the points $a$ and $b$ in $\mathbb{H}_2$, [5.5]. Imagine a particle that travels along different routes over the surface of the ...
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Shortest helical path with bumps on a cylindrical surface : 3D Convex 'Line' instead of hull?

I am studying the effect of fiber overlap during filament winding on a cylindrical surface. Every time a fiber crosses over another fiber, due to the thickness of the fiber under it, it 'bridges' for ...
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Simple formulas for geodesics of the Poincaré disk?

Let $D$ be the Poincaré disk endowed with the metric $g=4\dfrac{dx^2+dy^2}{(1-x^2-y^2)^2}$. I want to find a sample equation for the geodesics between two points $p$ and $q$ in the disk. We know that ...
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Geodesic convexity of small balls in Alexandrov CBB spaces

If $X$ is a Riemannian manifold, it is known that, for $p\in X$, there is some $\epsilon>0$ such that $B(p,\epsilon)$ is geodesically convex. Geodesically convex means there is a minimizing ...
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When does the union of all geodesics equal the metric interval?

Definitions: Throughout let $(M,d)$ be a geodesic metric space [cf. p. 104 of EoD]* with $d$ the (strictly) intrinsic metric, i.e. for all $x,y \in M$, $d(x,y)$ equals the length of any minimizing ...
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The exponential map is generally not holomorphic.

Prove that $\exp_0:\mathbb{C}\to \mathbb{D}$ is not holomorphic, where $\mathbb{D}$ is the unit disk with hyperbolic metric $ds^2=\frac{4dzd\bar{z}}{(1-|z|^2)^2}$. I am reading Jost's Compact Riemann ...
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Is the intersection of quasi convex sets also quasi convex?

A subset $A$ of a geodesic space $X$ is called quasi convex if there exists a constant $k > 0$ so that if $x,y \in A$, the geodesic joining $x$ to $y$ is in the $k$ neighborhood of $A$. Is it true ...
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Can the geodesic equation be used to solve the Brachistochrone Problem?

Assume the initial condition is that a point mass starts at height $y_0$. After descending to height $y < y_0$, we know that its speed will be $v = \sqrt{2mg(y_0 - y)}$. Thus, the displacement ...
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Studying Geodesic Deviation and Tidal Forces

I have been studying general relativity from a mathematical point of view and I was wondering what would be a good material (lecture notes, books, and so on) to study geodesic deviation and tidal ...
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Are half spaces in Hadmard manifold geodesically convex?

Given a Hadamard manifold $M$ (complete, simply connected and of nonpositive curvature) and two points $x,y\in M$ I want to consider the half space $H(x,y)=\{z\in M\mid d(x,z)\leq d(z,y)\}$. I wonder ...
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How to solve the geodesic equation?

Given this metric: $$ g_{ij} = \begin{pmatrix} r^2& 0 \\ 0 & r^2 \sin^2 {\theta}\\ \end{pmatrix} $$ How can I calculate the Christoffel Symbols and then arrive to the geodesic equation? $$ \...
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Straight line on a mobius strip

I have a mobius strip that is twisted by 540° degrees (not only 180° as the usual mobius strip). Also it is one that can't be constructed with a paper-strip. I created it with this OpenSCAD code: <...
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1 answer
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Geodesic through a point joining other two points

Let $M$ be a closed (compact without boundary) Riemannian manifold. Let $p,q\in M$ and $a\in M$ such that $d(a,p)=d(a,q)$. Does there always exists a (smooth) geodesic joining $p$ and $q$ passing ...
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If I were to stand on a flat torus, would I be able to see myself from behind?

The title pretty much says it all : from any point on a flat torus, there is at least one geodesic from that point that goes back to it. And, by definition of a flat torus, that geodesic is a line. So,...
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distance function on compact Riemannian manifold

Show that: Let $(M,g)$ be a compact Riemannian manifold, then the distance function $r(x) := \operatorname{dist}(x, p)$ can not be smooth on $M$\{$p$}. (1)As i know, by Hopf-Rinow theorem, $M$ must be ...
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Geodesic traingle is contained within the $3\delta$ ball centered at any of its vertex

Given any vertex of a geodesic triangle in hyperbolic space, show that the entire triangle is contained in the $3\delta$ ball centred at that vertex, where $\delta$ is the constant of hyperbolicity. ...
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Proper Way to view vector field component functions with respect to covariant derivative

I follow this lecture https://www.youtube.com/watch?v=2eVWUdcI2ho&list=PLFeEvEPtX_0S6vxxiiNPrJbLu9aK1UVC_&index=8 and at minute 35, the covariant derivative is used to calculate the geodesic ...
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Space form of c=-1

My problem relates to space form and geodesics in Riemman Geometry.Let $D=${$ x^ {2} + y^ {2} \leqslant 1$} with Poincare metric $g=\frac{4}{(1-x^ {2}-y^{2})^{2}}(dxdx+dydy)$ (1)Show that D is space ...
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Exponential map of a geodesic

I am working on geodesics with the book of Abeta M. and Tovena. F. There they defined in Definition 5.2.3 the notion of exponential map: The exponential map $exp_p:\mathcal{E}_p \to S$ at $p$ is ...
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Is my proof that a constant speed geodesic is a geodesic correct?

Given a metric space $(X,d)$ the following is a definition of a geodesic from the book of Santambrogio. I want to show that a constant speed geodesic is a geodesic : is the following enough? Note ...
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Is there a formal term for the set of all (geodesic) curves passing through a point?

My context is the spacetime of general relativity. I want to describe the fact that every spacetime event is the intersection of an infinite number of world-lines. I found myself reaching for a term ...
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Asymptotic bi-infinite geodesics in CAT($-1$) space coincides

Suppose $\gamma_1,\gamma_2:(-\infty,\infty)\to (X,d)$ be two bi-infinite geodesic such that $d(\gamma_1(t),\gamma_2(t))<K$ for all $t$. Here $(X,d)$ is a CAT$(-1)$ space. Then Image($\gamma_1$)=...
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Set geodesic convexity on positive curvature manifolds under linear transformation

Let M be a Riemannian manifold in $\mathbb{R}^n$ of positive sectional curvature and S be a geodesically convex set on M and $T \in GL(n, \mathbb{R})$. Is it true that $T(S)$ is geodesically convex in ...
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Every geodesic has constant speed, but why not also the other way around?

Take a (smooth) manifold $M$, and a curve $c : I \rightarrow M$, where $I$ is an interval. Then $c$ is a geodesic, if $c' : I \rightarrow \mathbb{R}^n$ is a parallel vectorfield along $c$, i.e. the ...
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Is the hyperbolic plane convex?

I'm attending a lecture series about introduction to non-Euclidian Geometry, but it is focused on the intuition of that topic without giving me the tools to analize the following question: Is the ...
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Is there a name for connections with this property?

A little preamble: an affine connection on the tangent bundle of a Riemannian (or pseudo-Riemannian) manifold is compatible with the metric if the covariant derivative of the metric is zero. Of course,...
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$A \in O_+(1, n)$ with a given plane of fixed points

I've been trying to understand geodesics of hyperbolic space $H^n$, and found a very similar question and a helpful answer here: Geodesics and Distance in Hyperbolic Space However, I found it ...
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Local geodesics

Suppose I have a $K$-local geodesic in a metric space $M$, $\alpha(t) : [0,L] \rightarrow M$, meaning $$\forall t ,t' \in [0,L] \quad \text{s.t.} \quad |t-t'|\leq K, \quad |\alpha(t)-\alpha(t')|=|t-t'|...
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1 answer
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How can I find the geodesics on the cylinder $x^2 + y^2 = 1$?

Definition Let's say we have a manifold $M$, and a curve $\gamma : \: I \ni t \rightarrow \gamma(t) \in M$ We call that curve a geodesic $\Leftrightarrow$ the vector field consisting of $\dot{\gamma}(...
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Geodesics as fixed sets of isometries [duplicate]

I have an issue with an exercise concerning Geodesics as fixed sets of isometries. Let $M$ be a Riemannian manifold, $\gamma:I\to M$ be a curve which is parametrized with constant speed, and $f:M\to ...
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