Geodesic connectedness of semi-Riemannian manifolds

Sánchez, Miguel

doi:10.1016/s0362-546x(01)00427-8

Cited by 32 publications

(27 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that there are several ways to define what it means for ∂M to be convex with respect to a Riemannian metric on M . Different viewpoints and the relations between them are surveyed in [21]. Perhaps, the most common way is to deem ∂M convex with respect to a Riemannian metric on M if and only if the second fundamental form of ∂M with respect to this metric is nonnegative-definite on ∂M .…”

Section: Formulation Of the Existence Resultsmentioning

confidence: 99%

Quasilinear parabolic equations and the Ricci flow on manifolds with boundary

Pulemotov

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

show abstract

Section: Formulation Of the Existence Resultsmentioning

confidence: 99%

Quasilinear parabolic equations and the Ricci flow on manifolds with boundary

Pulemotov

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

show abstract

“…There are also different ways to prove that, for a complete M (or equivalently D), the boundary ∂D is convex if and only if the domain D is convex (see the review [27]).…”

Section: Preliminariesmentioning

confidence: 99%

Convex domains of Finsler and Riemannian manifolds

et al. 2010

Self Cite

View full text Add to dashboard Cite

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.Comment: 22 pages, AMSLaTex. Typos corrected, references update

show abstract

“…One can also wonder for the connectedness of x 0 , x 1 by means of a geodesic even if they are not causally related, as in variational frameworks described below. Although this question has a geometrical interest (see for instance the survey [37]), it does not have a direct physical interpretation, nor equivalence for LFE. …”

Section: Connectedness Through Solutions To the Lfementioning

confidence: 99%

Connecting Solutions of the Lorentz Force Equation do Exist

Minguzzi

Sánchez

2006

Commun. Math. Phys.

View full text Add to dashboard Cite

Recent results on the maximization of the charged-particle action I x0,x1 in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I x0,x1 over a given causal homotopy class C of curves connecting two causally related events x 0 ≤ x 1 . Action I x0,x1 is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class C. Moreover, the maximum σ 0 on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C 2 ). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x 0 ≪ x 1 can be connected by means of a timelike solution of the Lorentz force equation (LFE) corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m) in the nonexact case.As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.

show abstract

Geodesic connectedness of semi-Riemannian manifolds

Cited by 32 publications

References 40 publications

Quasilinear parabolic equations and the Ricci flow on manifolds with boundary

Quasilinear parabolic equations and the Ricci flow on manifolds with boundary

Convex domains of Finsler and Riemannian manifolds

Connecting Solutions of the Lorentz Force Equation do Exist

Contact Info

Product

Resources

About