# Tagged Questions

216 views

### Are all null curves of a Lorentzian metric extrema?

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of $$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$ The so-called "null curves" are ...
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### Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem  ...
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### Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. Some background may be ...
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### For what metrics are circles solutions of the isoperimetric problem?

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...
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### Reference for a dual isoperimetric problem and solution

I am trying to track down the first published solution to the following problem: What curves within the unit disc in the plane and endpoints on the unit circle, minimize their length (within the ...
Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H$ is some prescribed function (small, smooth, vanishing near the endpoints to any ...