Geodesics and Jacobi fields of pseudo-Finsler manifolds

Javaloyes, Miguel Ángel; Soares, Bruno Learth

doi:10.5486/pmd.2015.7028

Cited by 22 publications

(28 citation statements)

References 22 publications

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“…The proof follows by using the same arguments as in the semi-Riemannian case (see [27,Proposition 9.30] for instance), and taking into account that the Jacobi fields are well-enough behaved in the Finslerian settings (see [18,Section 3.4], especially Proposition 3.13 and Lemma 3.14). With this previous result at hand, we are ready to prove the following characterization: Proof.…”

Section: Killing and Conformal Fields For Finsler Manifoldsmentioning

confidence: 99%

On a monodromy theorem for sheaves of local fields and applications

Herrera

Javaloyes

Piccione

2016

RACSAM

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Abstract. We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. In particular, in the case of admissible regular sheaves of local fields defined on a simply connected manifold, we obtain a global extension result for every local field of the sheaf. This generalizes previous works of Nomizu [26] for semi-Riemannian Killing fields, of for conformal fields, and of Amores [1] for fields preserving a G-structure of finite type. The result applies to Finsler or pseudo-Finsler Killing fields and, more generally, to affine fields of a spray. Some applications are discussed.

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Section: Killing and Conformal Fields For Finsler Manifoldsmentioning

confidence: 99%

On a monodromy theorem for sheaves of local fields and applications

Herrera

Javaloyes

Piccione

2016

RACSAM

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“…The results and methods in this paper may be extended toward different directions, for instance, pseudo-Finsler manifolds considered in [29,30,31], and the Lagrange geometry introduced by J. Kern [32] and developed by R. Miron and M. Anastasiei [48].…”

Section: Discussionmentioning

confidence: 99%

Splitting lemmas for the Finsler energy functional on the space ofH¹-curves

2016

Proc. London Math. Soc.

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We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of H1‐curves around a critical point or a critical double-struckR1 orbit of a Finsler isometry‐invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll–Meyer's splitting lemmas on Riemannian manifolds [Gromoll and Meyer, ‘On differentiable functions with isolated critical points’, Topology 8 (1969) 361–369; Gromoll and Meyer ‘Periodic geodesics on compact Riemannian manifolds’, J. Differential Geom. 3 (1969) 493–510]. As an application, we extend to Finsler manifolds a result by Grove and Tanaka [‘On the number of invariant closed geodesics’, Acta Math. 140 (1978) 33–48; Tanaka, ‘On the existence of infinitely many isometry‐invariant geodesics’, J. Differential Geom. 17 (1982) 171–184] about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.

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“…The idea of the proof is based on [2, Theorem 5.63]. Let us briefly recall it, accepting results on Jacobi field on Finsler spaces; see [7] and [10]. For p ∈ f −1 (c), consider the geodesic t → γ p (t) = exp p (tξ).…”

Section: Question 12 and Proof Of Theorem 13mentioning

confidence: 99%

On Finsler transnormal functions

Alexandrino

Alves²,

Dehkordi

2019

Differential Geometry and its Applications

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In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition, when the function is defined on a compact analytic manifold M . d c 1

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Geodesics and Jacobi fields of pseudo-Finsler manifolds

Cited by 22 publications

References 22 publications

On a monodromy theorem for sheaves of local fields and applications

On a monodromy theorem for sheaves of local fields and applications

Splitting lemmas for the Finsler energy functional on the space ofH¹-curves

On Finsler transnormal functions

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Geodesics and Jacobi fields of pseudo-Finsler manifolds

Cited by 22 publications

References 22 publications

On a monodromy theorem for sheaves of local fields and applications

On a monodromy theorem for sheaves of local fields and applications

Splitting lemmas for the Finsler energy functional on the space ofH1-curves

On Finsler transnormal functions

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Splitting lemmas for the Finsler energy functional on the space ofH¹-curves