On a Class of Geodesically Connected Lorentzian Manifolds

✍ Scribed by Flavia Antonacci; Rosella Sampalmieri

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 345 KB
• Volume: 138
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1997.3253

✦ Synopsis

In this paper we study the geodesical connectedness of Lorentzian manifolds. We consider a connected manifold M=M 0 _R, where M 0 is a complete Riemannian manifold endowed with a Lorentzian metric g of splitting type. We prove that, under suitable hypotheses on the coefficients of the metric g, M is geodesically connected.

1997 Academic Press

1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS

In this paper we study the geodesical connectedness of the Lorentzian manifolds of splitting type.

We recall that a Lorentzian (or Riemannian) manifold (M, g) is said geodesically connected if every couple of its points can be joined by a geodesic.

While for Riemannian manifolds the Hopf Rinow theorem [7] gives a satisfactory answer to this problem, in the Lorentzian case, not even the compactness assumption on the manifold is sufficient to guarantee the geodesical connectedness (see [5]).

Results in this context are provided, among the others, by [2 5, 10], using variational methods. In this paper, by applying techniques similar to those used in [1,6,10], we improve the result of [4], on the asymptotic article no. DE963253