Closed Geodesics with Lipschitz Obstacle

Degiovanni, Marco; Morbini, Laura

doi:10.1006/jmaa.1999.6342

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…To complete the our study about geodesics, we want to compare our approach with Degiovanni-Marzocchi-Morbini one. These authors in [12,18] study the problem of finding geodesics in manifolds with boundary by using the weak slope theory; under certain hypothesis on the conical manifold, both approaches are possible. We'll see that there is a strong bound between these theories, and we'll be able to recover a regularity result.…”

Section: Comparison With Weak-slope Approachmentioning

confidence: 99%

“…Furthermore, the geodesic problem on a singular manifold is studied by Degiovanni and Morbini, in [12]. In section 5 we will point out the common feature and the main differences between our approach and the Degiovanni and Morbini one.…”

Section: Introductionmentioning

confidence: 97%

“…we show some examples in which an high multiplicity, or a 0 multiplicity of a geodesic occurs. Also, the definition of index that we give, allows us to make a comparison between our approach to conical manifold and the approach of Degiovanni, Marzocchi and Morbini (Section 5), which is based on the concept of the weak slope (see [7,9,10,11]), and was presented in [12,18]…”

Section: Introductionmentioning

confidence: 99%

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Morse Theory for Geodesics in Conical Manifolds

Ghimenti

2007

MedJM

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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 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. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided

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Section: Comparison With Weak-slope Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Morse Theory for Geodesics in Conical Manifolds

Ghimenti

2007

MedJM

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“…These results essentially deal with a nondegenerate situation (in a suitable sense); under appropriate assumptions, also a degenerate situation is treated in [18], through a nonsmooth adaptation of the Generalized Morse Lemma of [12] in a neighbourhood of an isolated critical point. In [18], the critical groups are defined via Alexander-Spanier cohomology, since use is made, as a substitute for Theorem 4, of [11,Theorem 2.7]. The latter result shows that, letting K a be any set in Theorem 4, the inclusion f a ⊂ f b still induces an isomorphism H * (f a ) → H * (f b )…”

Section: The Connection Between the (Ps) Condition And Theorem 3 Is Ementioning

confidence: 99%

On the second deformation lemma

Corvellec¹

2001

TMNA

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“…The weak slope was introduced by Marco Degiovanni and Marco Marzocchi in [DM94] (see also [Deg97,CD95,CDM93]). Moreover we refer to [DM99,MM02] for a weak slope approach to geodesic problem and to [Ghi04] for a detailed comparison with our approach.…”

Section: Introduction and Basic Definitionmentioning

confidence: 99%

Geodesics in conical manifolds

Ghimenti¹

2005

TMNA

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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.

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Closed Geodesics with Lipschitz Obstacle

Abstract: Geodesics on a general subset M of ‫ޒ‬ n are considered. When M is the closure of an open subset with Lipschitz boundary and suitable topological conditions are fulfilled, the existence of a nonconstant closed geodesic is proved. ᮊ 1999 Academic Press

Cited by 7 publications

References 14 publications

Morse Theory for Geodesics in Conical Manifolds

Morse Theory for Geodesics in Conical Manifolds

On the second deformation lemma

Geodesics in conical manifolds

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