Geodesics in conical manifolds

Abstract

The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of 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) with variational geodesics, i.e. critical points of the energy functional. We prove a deformation lemma (Theorem 2) which leads us to extend the Lusternik-Schnirelmann theory to conical manifolds, and to estimate the number of geodesics (Theorem 10 and Corollary 10.1). 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 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.

Morse Theory for Geodesics in Conical Manifolds

“…In this section we present some peculiarities of the study of geodesics in conical manifold, which motivate Definition 1.2. Also, we resume briefly some result contained in [15] which will be useful for this paper.…”

“…By this definition, we were able to prove a deformation lemma that allows us to apply the main theorems of the calculus of variation in the large. In particular we can estimate the number of geodesics using the Ljusternik-Schnirelmann theory ( [15,Theorem 1]). In this work we will associate to each geodesic γ an index i(γ) similar to the Morse index, that allows us to obtain some information on the qualitative properties of the geodesics.…”

“…In a previous work [15] we introduced conical manifolds as submanifolds of R n , giving the following definition: Definition 1.1. A conical manifold M is a complete n-dimensional C 0 sub manifold of R m which is everywhere smooth, except for a finite set of points V .…”

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The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of R n with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work.

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Evolution of clinical and environmental health applications of exhaled breath research: Review of methods and instrumentation for gas-phase, condensate, and aerosols

“…SEV is metabolized to hexafluoroisopropanol (HFIP), which is eliminated in exhaled breath. 271 Potentially liver toxic compounds of interest can be added to cellular in vitro systems in the presence of MTBE or SEV to see if production of TBA or HFIP, respectively, is disrupted by the presence of the test compound. 262,271…”