On completeness of certain families of semi-Riemannian manifolds
Semi-Riemannian manifolds with a suitable set of conformal symmetries are shown to be complete. Locally warped products are studied and warped-completeness is introduced. In the case of definite and complete basis, several assumptions on the growth of the warping function yield some of the three kinds of completeness. The case of 1-dimensional basis (including a known family of relativistic space-times) is specially studied. Null warped-completeness is related to the completeness of a certain conformal metric on the basis. Several examples and counter-examples explaining the main results are also given.
Mathematics Subject Classifications (1991)53C22 53C50
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- 1.Beem, J. K. and Ehrlich, P. E.:Global Lorentzian Geometry. Pure and Applied Math., Marcel Dekker, 1981.Google Scholar
- 2.Carrière, Y.: Author de la conjecture de L. Markus sur les variétés affines,Invent. Math. 95 (1989), 615–628.Google Scholar
- 3.Furness, P. M. D. and Fedida, E.: Sur les structures pseudo-riemannienes plates de variétés compactes,J. Nigerian Math. Soc. 5 (1986), 63–78.Google Scholar
- 4.Kobayashi, S. and Nomizu, K.:Foundations of Differential Geometry, Vol. I, Wiley Interscience, 1963.Google Scholar
- 5.Lafuente, J.: A geodesic completeness theorem for locally symmetric Lorentz manifolds,Rev. Mat. Univ. Complutense Madrid 1 (1988), 101–110.Google Scholar
- 6.Marsden, J.: On completeness of homogeneous pseudo-Riemannian manifolds,Indiana Univ. Math. 22 (1973), 1065–1066.Google Scholar
- 7.O'Neill, N.:Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983.Google Scholar
- 8.Romero, A. and Sánchez, M.: On completeness of geodesics obtained as a limit,J. Math. Phys. 34(8), 3768–3774.Google Scholar
- 9.Yurtsever, U.: A simple proof of geodesical completeness for compact space-times of zero curvature,J. Math. Phys. 33 (4) (1992), 1295–1300.Google Scholar