Mathematics Stack Exchange
Mathematics 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.

A geodesic is a generalization of a straight line to curved space. It is a length-minimizing curve, which is equivalent to a path that a particle which is not accelerating would follow.

On a Riemannian manifold, a geodesic coordinatized by coordinates $x^k$ satisfies the ordinary differential equation known as the geodesic equation: $$\frac{d^2x ^k}{dt^2}+\sum_{ij}\Gamma^{ij}_k\frac{d x^i}{dt}\frac{dx^j}{dt}=0.$$

In general relativity, a geodesic will describe the motion of point particles under the influence of gravity.

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Where to build a bridge to cross a river in the shape of an annulus

There is a river in the shape of an annulus. Outside the annulus there is town "A" and inside there is town "B". One must build a bridge towards the center of the annulus such that the path from A to B crossing the bridge is the shortest possible.…
João Rimu
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Can every curve on a Riemannian manifold be interpreted as a geodesic of a given metric?

Given a metric $g_{\mu\nu}$ it is possible to find the equations of the geodesic on the Riemannian manifold $M$ defined by the metric itself: $$\frac{d^2x^a}{ds^2} + \Gamma^{a}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$ where: $$\Gamma^a_{bc} =…
Riccardo.Alestra
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St. Basil's cathedral, Moscow steeple shape

Onion-shaped dome cathedral architecture seen here appears to include in its lower part a geometry of positive, and in upper (steeple) part negative Gauss curvature. The corresponding elliptic and hyperbolic geodesics transition at an inflection…
Narasimham
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Why are we interested in closed geodesics?

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics representing some non trivial homotopy class of closed…
Lor
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How can we find geodesics on a one sheet hyperboloid?

I am looking at the following exercise: Describe four different geodesics on the hyperboloid of one sheet $$x^2+y^2-z^2=1$$ passing through the point $(1, 0, 0)$. $$$$ We have that a curve $\gamma$ on a surface $S$ is called a geodesic if…
Mary Star
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Exponential map on the $n$-sphere

I might need some help on the following exercise : Let $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$ be the unit $n$-sphere. For any $p \in \mathbb{S}^{n}$, we have $$T_{p}\mathbb{S}^{n} = p^{\perp} = \lbrace v \in \mathbb{R}^{n+1}, \, p \cdot v = 0…
pitchounet
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Hamiltonian for Geodesic Flow

I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where $$H = \frac{1}{2}g^{ij}p_i p_j$$ but I am stuck. Could somebody show me how to complete the calculation, or where I've…
Edward Hughes
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Metal Ball Cage Template Cardinality: A Brilliantly Lazy PROOF

N.B. - I'm looking for the simplest way to ascertain the number of templates $T$ (see below) comprising the structure from just one angle alone; that is, I'm sitting down looking up at this thing, and I want a way to compute its cardinality based…
Trancot
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Expression of the Hyperbolic Distance in the Poincaré Upper Half Plane

While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia. In the page Poincaré Half Plane Model it is…
Giovanni De Gaetano
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Geodesic between two points

I have a question about geodesics. So far I know that for any surface $S$ defined by some immersion $f: U \subset\mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3,$ we have that for any point on the surface and any direction, there exists locally a…
user66906
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geodesic computation: "energy" minimization versus arc length minimization

Is it true that applying the Euler-Lagrange equation to the integral $E(\gamma)=\int_{t_1}^{t_2} g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'\operatorname{d}\!t$ rather than the arc length integral $L(\gamma)=\int_{t_1}^{t_2}…
J. Heller
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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…
Asaf Shachar
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Geodesics of Sasaki metric

I would like to ask the community for a reference on the following question: Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that the orbits of the geodesic flow…
matgaio
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What does it mean "being geodesic" is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. My question is Let $\alpha(s)$ be a geodesic on a…
Semsem
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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 is a geodesically convex geodesic ball $\mathcal…
Sophie
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