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J. J. Pareja, Class. Quantum Grav. 16 (1999) 3823. This page intentionally left blank NEW DIRECTIONS FOR THE NEW MILLENNIUM Frederick J. Ernst * FJE Enterprises. Potsdam, New York 13676, USA Keywords: Exact solutions, potentials formaliosm, Geroch group. 1. INTRODUCTION As we stand at the threshold of a new millennium, perhaps it would not be amiss to reflect upon what has been accomplished since Einstein presented general relativity to the world, to ponder where we should attempt to go from here, and to speculate about what will actually take place in the future.

MAXIMALLY SYMMETRIC TOTALLY GEODESIC HYPERSURFACE Applying this theory to the maximally symmetric space (47), with the identification the normal to unit vector and are given by which, substituted in (60), imply these conditions are identically fulfilled because of the structure of the considered metric (47). The substitution in yields Therefore, a space hypersurface iff : admits a totally geodesic maximally symmetric Consequently, the metric of mally symmetric hypersurface admitting a totally geodesic maxiis reducible to Some properties of a totally geodesic hypersurface are: the lines of curvature of a totally geodesic hypersurface are indeterminate, and the normals to a totally geodesic hypersurface are parallel in the enveloping space.

J. Ernst, preprint gr-qc/0010058. [10] H. J. K. Metzner, Proc. Roy. Soc. (London) A269 (1962) 21. K. Sachs, Proc. Roy. Soc. (London) A270 (1962) 103. T. Bishop, R. Gomez, L. Lehner, M. Maharaj and J. Winicour, Phys. Rev. D56 (1997) 6298. Preprint gr-qc/9705033. This page intentionally left blank ON MAXIMALLY SYMMETRIC AND TOTALLY GEODESIC SPACES Alberto Garcia* Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN Apdo. Postal 14-740, 07000 México DF, México Abstract The purpose of this note is the demonstration of the following theorem: the metric of an (n+1)–dimensional space admitting a totally geodesic maximally symmetric hypersurface is reducible to Keywords: Geodesic spaces, maximally symmetric spaces, exact solutions.

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