# 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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### 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 ...
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### 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 ...
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### 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." ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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: ...
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### 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 ...