Geodesics in conical manifolds (Q2568590)

The paper extends the definition of geodesics to conical manifolds, defined as submanifolds of \(\mathbb R^n\) with a finite number of singularities. The author develops a suitable approach both for the local geodesic problem and for the calculus of variation in the large. First a definition is given, which links the local solutions of the Cauchy problem with variational geodesics, i.e. critical points of the energy functional. Further, a deformation lemma permits to extend the Lusternik-Schnirelmann theory to conical manifolds, and to estimate the number of geodesics. As well are provided applications in which conical manifolds arise naturally; in particular, a special place has the brachistochrone problem for a frictionless particle moving in \(S^n\) or in \(\mathbb R^n\) in the presence of a potential unbounded from below. An Appendix presents the main results in a general framework.