Articles of geodesic

Exponential map of Beltrami-Klein model of hyperbolic geometry

In the Betrami-Klein model of hyperbolic geometry, geodesics are represented as straight lines. Hence the exponential map of a tangent vector $\mathbf{v}$ at a point $\mathbf{p}$ is $\mathbf{p} + \lambda \mathbf{v}$, where $\lambda$ is a scalar that depends on $\mathbf{p}$ and $\mathbf{v}$. For example, suppose $\mathbf{p} = 0$. Then the exponential map is $$ \exp_\mathbf{p}(\mathbf{v}) […]

Distance between two points on the Clifford torus

How can I obtain the distance between two points $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and $\mathbf{y}=(y_1,y_2,y_3,y_4)$ that belong to the $2$-torus $\mathbb{S}^1\times \mathbb{S}^1$? This is, I want to measure the distance (I do not require the geodesic) of $\mathbf{x}$ to $\mathbf{y}$ along the manifold $$\mathbb{S}^1\times\mathbb{S}^1=\big\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=1,x_3^2+x_4^2=1\big\}.$$ I guess that there should be a way of doing this considering an isomorphism(?) […]

On continuously uniquely geodesic space II

This question was inspired by this answer of @wspin. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question : Is a complete uniquely geodesic space, continuously uniquely geodesic ?

Distance function on complete Riemannian manifold.

Let $\left(M, g\right)$ be a complete Riemannian manifold. Let us fix a point $p \in M$ and consider the distance function $$ r(x) := \operatorname{dist}(x, p). $$ I would like to characterize the point where $r$ is not smooth. Is it true that they must be critical points of the function $\exp_p$? And is it […]

On continuously uniquely geodesic space

This question was inspired by this comment of @68316. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question : Is there a uniquely geodesic space which is not continuously uniquely geodesic ?

Geodesics on $S^2$

Correct description of this problem is here: Geodesics on $S^2$ with specific Riemannian metric I was asked to show the following: Given a Riemannian metric on $S^2$, let $x \in S^2$ be the north pole, and let $V(\theta) \in T_xS^2$ be unit tangent vectors at $x$. Then there exists a constant $t > 0$ such […]

What exactly is the geodesic flow?

I understand what a geodesic is, but I’m struggling to understand the meaning of the geodesic flow (as defined e.g. by Do Carmo, Riemannian Geometry, page 63). I can state my confusion in two different ways: 1) Do Carmo writes: Why does a geodesic $\gamma$ uniquely define a vector field on an open subset? In […]

Calculation mistake in variation of length functional?

This should be pretty simple to check if you know the basics of variational calculus. I feel like I am making an obvious mistake somewhere like not using chain rule somewhere. Let $g : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth metric on some domain in $\mathbb{R}^n$, and let $\gamma : [0,1] \rightarrow \mathbb{R}^n$ […]

Example for conjugate points with only one connecting geodesic

$\newcommand{\ga}{\gamma}$ $\newcommand{\al}{\alpha}$ I would like to find an example for a Riemannian manifold, that has two conjugate points $p,q$ with only one connecting geodesic between them. (This is the geodesic they are conjugate along) Explanation: Consider a parametrized family of geodesics starting from a fixed point $p$, i.e: $\ga_s(t)=\ga(t,s), \ga_s(0)=\ga_0(0)=p$ where for each fixed $s$ […]

Why is the geometric locus of points equidistant to two other points in a two-dimensional Riemannian manifold a geodesic?

Let $M$ be a 2-dimensional Riemannian manifold, $x,y \in M$. Why is the set of points $\{z | d(z,x) = d(z,y)\}$ a geodesic? What can we say about higher-dimensional Riemannian manifolds?