Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
0answers
21 views

Triangle Inequality on geodesic surfaces

I am not well-versed in geometry relating to manifolds so pardon my lack of correct terminology. Simply put, if I have some geometric surface (that can be found in the real physical world, such as ...
2
votes
1answer
56 views

Existence of conjugate points along geodesics on a complete surface with curvature bounded below

I'm introducing myself to Jacobi fields and conjugate points along geodesics by studying Chapter 5 of Do Carmo's "Differential Geometry of Curves and Surfaces". I was trying this exercise (exercise ...
2
votes
0answers
28 views

Complete Riemannian manifolds and lengths of curves diverging to infinity [duplicate]

I've been working on the following exercise and got stuck: "Prove that a connected Riemannian manifold is complete if and only if every regular curve that diverges to infinity has infinite length." ...
1
vote
1answer
54 views

Example of the Manifold with a Non-Surjective Exponential Map at a Point

I need to find an example of a connected riemannian manifold $(M,g)$ and a point $p \in M$ such that the exponential map $\exp_p : T_pM \to M$ is well-defined, but is not surjective. Taking $\mathbb{...
1
vote
1answer
38 views

Show γ(t) = (a cos(t), a sin(t), t) is a geodesic of the cylinder .

The first part of this question required me to compute the general geodesic of a surface of revolution. And I obtained a general equation using Christoffel's symbols. The second part asked me to find ...
0
votes
0answers
16 views

Geodesic distance from point to line

Is there a equation that finds the geodesic distance from a point to a line, on a generic surface? For example, if I know two arbitary triangles $T_0, T_1$, with its corresponding 3D points, and ...
2
votes
0answers
30 views

Realizing every rotation of a tangent space on a sphere as a parallel transport

I am taking a course on elementary differential geometry, in which we use Do Carmo "Differential Geometry of Curves and Surfaces" as our textbook. I have handed in a written assignment solving - well, ...
2
votes
1answer
36 views

Geodesics of the right circular cone

I have to find teh geodesics of the cone: $$C={(x,y,z)\in R^3:x^2+y^2=z^2,z>0}$$ My idea is use that the geodesic curvature must be 0 for be a geodesic.Then I use the parametrization: $$X(u,v)=(...
1
vote
0answers
21 views

Derivation of the Geodesic Equation from Hamilton-Jacobi Theory

I'm trying to prove a result in my general relativity class and I'm confused. If we have the Hamiltonian: $$ H = g^{\mu\nu}p_\mu p_\nu + m^2 $$ Subject to Hamilton's Equations: $$\frac{\partial x^\mu}{...
0
votes
0answers
23 views

How to calculate the geodesic curvature of a discrete 3D curve?

I have coordinates of a set of points that form a closed loop that lies in a 3D surface. I know the equation of the surface and I can calculate it's surface normal at any point. I found that for a ...
2
votes
0answers
44 views

Geodesics under coordinate transformation

Consider components of metric tensor $g'$ in a coordinate system $$g'= \begin{pmatrix} xy & 1 \\ 1 & xy \\ \end{pmatrix} $$ We can transformation rule which brings $g'$ to euclidean metric $...
0
votes
0answers
15 views

infinitely many distinct hyperbolic geodesics.

Can you prove this? Consider the geodesic $L = \{it: t \in R, t>0\}$ in $H^2$, and consider the point $w = i+1$ which is not on $L$. Show that there are infinitely many distinct hyperbolic ...
2
votes
1answer
32 views

Geodesics on a Riemannian manifold under non-Levi-Civita connections

I'm a beginner on this topic—so please comment if anything is ambiguous, unclear, or wrong. In particular, I'm trying to figure out how to think of geodesics under arbitrary connections. Background ...
7
votes
2answers
319 views

Example of compact Riemannian manifold with only one closed geodesic.

The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic. Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is ...
0
votes
1answer
16 views

Symmetrizing terms of Christoffel symbols

In equation 8 in this paper they claim that in the geodesic equations this can be done: $ \ddot{x}^\mu = \Gamma^\mu_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta = - \frac{1}{2} g^{\mu\nu}\left( \...
1
vote
0answers
36 views

Question about a proof on Spivak's Comprehensive Introduction to Differential Geometry

On the Addedum to the Chapter VI, vol. II, the first proposition states that to connections have the same geodesics if and only if their difference tensor is antisymmetric. When proving that if the ...
1
vote
0answers
16 views

Existence of timelike curve in a particular set up.

I encountered online the following exercise: Let $M$ be a Lorentz manifold of $\dim(M)=n$, and let $\psi:\Sigma\to M$ be a spacelike submanifold of dimension $n-2$ embedded into $M$. We will identify ...
1
vote
1answer
20 views

Distance on the sphere is convex

Let $\mathbb{S}^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:\mathbb{S}^n \to [0, \infty)$ defined by $d(p) = d(p,p_0) = \cos^{-1}( \langle p, p_0 \rangle)$. It ...
0
votes
1answer
50 views

Is a curve whose derivative has constant length a geodesic?

Given a (complete) riemannian Manifold $(M,g)$ and a curve $\gamma \colon [0,1] \to M$ with $$g\big( \dot{\gamma} (t) , \dot{\gamma}(t) \big) = C \quad \forall t \in [0,1]$$ where $C \neq 0$ is some ...
0
votes
3answers
47 views

Explanation for why $\mathbb R^2 \setminus \{0\}$ with the Euclidean metric is not a geodesic metric space?

A geodesic metric space is a metric space $X$ where for any two points $x, y \in X$ there exists a geodesic segment, i.e., an isometry $\gamma: [a,b] \to X$ where $\gamma(a)=x$ and $\gamma(b)=y$. I ...
5
votes
0answers
98 views

Lipschitz constant of the exponential map

Let $M$ be a smooth Riemannian manifold and let $p \in M$. Suppose $r \ll \text{inj}(p)$ (the injectivity radius at $p$) and fix $t \in (0,r)$ then define the map $$ T_pM \ni v \mapsto \exp_p(tv) \in ...
2
votes
3answers
57 views

Given two points with 3D coordinates, and three angle observations, how to caculate the coordinate of the third point

Given two points, $M_1(X_1,Y_1,Z_1)$ and $M_2(X_2,Y_2,Z_2)$,$P(X_p, Y_p,Z_p)$ is the unknown point. How to get the coordinates of $P$ by three angle observation . The picture below displays the ...
1
vote
0answers
79 views

Calculus of variation and isoperimetric problem with differential forms and moving frames

This question follows this one. I want to apply the calculus of variation with differential forms to three classical problem: 1. arc-length minimizing curve (geodesics) 2. area-minimizing surfaces (...
0
votes
0answers
53 views

Geodesic equations in polar coordinates (Euclidean space)

I tried to derive the geodesics in polar coordinate system (which should be a straight line since the metric is still Euclidean), and arrived the same equations as in this question: How to calculate ...
0
votes
0answers
13 views

displacement that follow a geodesic in a spherical coordinate system

I've a spherical coordinate system $(r,\theta,\varphi)$ with: $r$ the radius, in the interval $[0, +\infty[$ $\theta$ the inclination, in the interval $[0, \pi]$ $\varphi$ the azimuth, in the ...
2
votes
2answers
64 views

Closed geodesic definition

A geodesic is defined to be the curve in the manifold which has the shortest path/length between two points. But when the curve is closed and the two points are the same then the curve length can ...
2
votes
0answers
52 views

Fubini Study Geodesics on $\Bbb {CP}^n$

I want to solve the geodesic equation on $\mathbb{CP}^n$ with the Fubini Study metric $$ g_{ij}=\frac{\left(1+{\mid z\mid}^2\right)\delta_{ij}-\bar{z}_iz_j}{\left(1+{\mid z\mid}^2\right)^2} $$ I have ...
0
votes
1answer
32 views

How to prove the existence of infinite geodesics that do not intersect in hyperbolic space

My given question is: In hyperbolic space, given a geodesic $L$ and a point $p$ not lying on $L$, show there is an infinite number of geodesics through $p$ which do not intersect $L$. The "model" ...
0
votes
2answers
38 views

If there are “arbitrarily efficient” paths in a given direction, is there always a “best” path in that direction?

$\newcommand{\ep}{\epsilon}$ $\newcommand{\al}{\alpha}$ $\newcommand{\be}{\beta}$ Let $M$ be a Riemannian manifold, $S \subset M,p \in M \setminus{S}$, and $v \in T_pM$ be a unit vector. Let $d_S$ be ...
0
votes
0answers
38 views

derivation of Jacobi field equation (from do Carmo's riemannian geometry)

I have been reading about Jacobi fields in do Carmo's book "Riemannian geometry", and I have a little question. Let $(M,g)$ be a riemannian manifold and let $p\in M$ and $v\in T_p M$ such that $\...
1
vote
0answers
49 views

Deriving the geodesic equations on a cone. Are these equations correct?

So I'd like to derive the geodesic equations of a cone which I call $\mathcal{C}$. I believe I've done this correctly but would like a second opinion. $\mathcal{C}$ can be described by taking the line ...
0
votes
0answers
25 views

Geodesics of the complex projective space

Is the complex projective space, a geodesic space? Is the complex projective space, a convex space? Let H be hyperplane of $\mathbb{C} P^n$, Is $\mathbb{C} P^n\setminus H,$ a bounded convex space?
2
votes
0answers
31 views

Geodesics of null surface and induced connection

Although this is a question related to GR this is purely a math question, so I think this is the right place to ask it. Let $(M,g)$ be a Lorentzian manifold. Let $\phi : \Sigma\to M$ be one null ...
3
votes
1answer
82 views

Uniqueness of geodesics that join two points in exponential neighbourhoods

I'm currently working through some notes on surfaces in $\mathbb{R}^3$ and their geodesics, with the following definitions (it's kind of lengthy, I will highlight the key parts): For a sufficiently ...
2
votes
0answers
45 views

Showing that a curve is a geodesic of a surface?

Let $\gamma(u): I \to \mathbb{R}^3$be a unit speed curve and let $\vec{b}(u)$ be its binormal vector. Consider the surface $S$ given by the surface patch $\sigma(u,v) = \gamma(u)+v\vec{b}(u)$. Show ...
-1
votes
1answer
56 views

Solving $yy''+ky'^2-k=0$

During some calculations about geodesics, after some simplification I found the following Cauchy problem: $$yy''+ky'^2-k=0$$ $$y(0)=1,y'(0)=c$$ with $c>1$ and $0<k<1$. How could I solve it?...
1
vote
0answers
31 views

Prove that a geodesic under certain hypothesis has a conjugate point

If $(M,g)$ is a riemannian manifold, and $N$ a submanifold in $M$ with $\sigma$ a geodesic normal to $P$ at $p=\sigma(0)$ under the hypotheses: -$H(\sigma'(0))=g\left(\sigma'(0)),\vec{H_p}\right)>...
0
votes
1answer
41 views

geodesics in hyperbolic space

Let $M$ be the Poincare ball model of the Hyperbolic space, and let $\zeta \in T_0M$. In my lecture notes it is claimed that $$c(t)=\tanh(\Vert \zeta \Vert t )\zeta/\Vert \zeta \Vert$$ is the geodesic ...
-1
votes
1answer
77 views

Calculate slant range between two GPS coordinates, including altitude

Given two GPS lat/lon/altitude coordinates of two aircraft, how do I compute the slant range (line of sight distance) between them?
2
votes
1answer
54 views

Calculate initial conditions to integrate null geodesic

Suppose, this is the line element of a spherically symmetric FLRW metric, $$ ds^2 = -[1 + 2ψ(t,x)]dt^2 + a^2(t) [1 - 2ψ(t,x)]dx^2 $$ and the geodesic equation is, $$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \...
0
votes
0answers
27 views

Geodesic and Riemannian distances are equivalent via exponential map

Exercise 5.40, page 134, in the book A Course in Differential Geometry (Graduate Studies in Mathematics) by Thierry Aubin asks to prove that given $X, Y \in \mathbb R^n$, for any $\varepsilon>0$, ...
1
vote
1answer
51 views

Are Jacobi Fields the geodesics of the tangent bundle?

Let $(M, g)$ be a Riemannian manifold. Wikipedia states that "Jacobi fields correspond to the geodesics on the tangent bundle". I'm trying to undrestand this statement. Curves $c : I \to TM$ ...
0
votes
1answer
38 views

How to derive the Euler Lagrange equation for geodesics?

In my book, it says a geodesic is associated to the functional $\int_0^l |\gamma'|^2$ , with a metric g. It then jumps to $\ddot{\gamma}^k + \Gamma^k_{ij}\dot{\gamma}^i\dot{\gamma}^j = 0$ where $\...
2
votes
0answers
102 views

What volume is enclosed by $k$ evenly-spaced, overlapping American footballs whose axes are diameters of a unit sphere?

Take $k \in \Bbb N$ intersecting American footballs and configure them inside a unit sphere such that each football touches two opposite ends of the sphere. Each of the shapes are spaced evenly apart. ...
1
vote
0answers
45 views

Deduce the curvature of the manifold by probing it with geodesics

Suppose in order to probe the curvature of a manifold in $\Bbb R^2$, you decide to shoot off geodesics from the four corners of the square. Consider a set of geodesics, $K,$ each symmetrical about ...
1
vote
1answer
30 views

Euler Lagrange and Geodesics

I’m trying to use the Euler Lagrange equations to derive the geodesic equations. I’ve assumed a lagrangian: $$ L = {1\over 2} g_{ij}\dot x^i \dot x^j $$ So one of the terms of the equation requires: ...
6
votes
3answers
195 views

Does parallel transport change the subspace?

Let $M$ be a Riemannian manifold and $N$ be an immersed submanifold. Take $\gamma$ a geodesic starting at and perpendicularly to $N$. Let $X(t)$ be a vector field along $\gamma$ such that $X(0) \in T_{...
0
votes
0answers
56 views

Solution of geodesic equations of spherical coordinates

I have found the solution of geodesic equations of $3$d spherical coordinates and got 3 second-order equations in $r$, $\theta$, and $\phi$, but I want to solve out that $2$nd order ODE to find out ...
1
vote
0answers
52 views

When do two different Jacobi Fields commute in the sense of the Lie Bracket?

Let $\mathcal{M}$ an riemannian submanifold of euclidean space, i.e. $\mathcal{M} \subset \mathbb{R}^n$. Equipped with the Levi-Civita- Connection. Additionally let $\mathbf{\gamma}^1$ and $\mathbf{\...
4
votes
1answer
46 views

When are cone geodesics planar

I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section. One asked whether if you 'unroll the cone' the conic section becomes a straight line ...