Geodesics in conical manifolds

Marco Ghimenti



The aim of this paper is to extend the definition of geodesics to
conical manifolds, defined as submanifolds of ${\mathbb R}^n$ with a
finite number of singularities. We look for an
approach suitable both for the local geodesic problem and for the
calculus of variation in the large. We give a definition
which links the local solutions of
the Cauchy problem (1.1) with variational
geodesics, i.e. critical points of the energy functional.
We prove a deformation lemma (Theorem 2.2)
which leads us to extend the Lusternik-Schnirelmann theory to
conical manifolds, and to estimate the number of geodesics
(Theorem 3.4 and Corollary 3.5).
In Section 4, we provide some
applications in which conical manifolds arise naturally: in
particular, we focus on the brachistochrone problem for a
frictionless particle moving in $S^n$ or in ${\mathbb R}^n$ in the presence of
a potential $U(x)$ unbounded from below. We conclude with an
appendix in which the main results are presented in a general framework.


Geodesics; nonsmooth critical point theory; nonsmooth manifolds

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