Abstract
In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime \(\overline M \) equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when \(\overline M \) is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when \(\overline M \) is a generalized Robertson-Walker spacetime.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alías, L. J. (1999). Geom. Dedicata 77, 297-304.
Alías, L. J., Brasil, Jr., and Colares, A. G. To appear in Proc. Edinburg Math. Soc.
Alías, L. J., and Montiel, S. (2002). Uniqueness of spacelike hypersurfaces with constant mean curvature in generalized Robertson-Walker spacetimes. Differential Geometry Valencia 2001 (World Scientific P.C.), 59-69.
Alías, L. J., Romero, A., and Sánchez, M. (1995). Gen. Rel. Grav. 27, 71-84.
Alías, L. J., Romero, A., and Sánchez, M. (1997). Nonlinear Analysis TMA 30, 655-661.
Alías, L. J., Romero, A., and Sánchez, M. (1997). Tôhoku Math. J. 49, 337-345.
Alías, L. J., Romero, A., and Sánchez, M. Spacelike hypersurfaces of constant mean curvature in spacetimes with symmetries. To appear in WODGA'98, Proceedings of the 7th Fall Workshop on Differential Geometry and its Applications.
Brasil, Jr., and Colares, A. G. (1999). Matematica Contemporânea 17, 99-136.
Calabi, E. (1970). Proc. Symp. Pure Math. 15, 223-230.
Cheng, S. Y., and Yau, S. T. (1976). Ann. of Math. 104, 407-419.
Choquet-Bruhat, Y., and York, J. (1980). The Cauchy Problem, In General Relativity and Gravitation A. Held (ed.), Plenum Press, New York.
Goddard, A. J. (1977). Math. Proc. Cambridge Phil. Soc. 82, 489-495.
Kamishima, Y. (1993). J. Differential Geom. 37, n 3, 569-601.
Lichnerowicz, A. (1944). J. Math. Pures Appl. 23, 37-63.
Marsden, J. E., and Tipler, F. J. (1980). Phys. Rep. 66, 109-139.
Montiel, S. (1988). Indiana Univ. Math. J. 37, 909-917.
Montiel, S. (1999). Math. Ann. 314, 529-553.
Rainer, M., and Schmidt, H.-J. (1995). Gen. Relativity Gravitation 27, n 12, 1265-1293.
Ramanathan, J. (1987). Indiana Univ. Math. J. 36, 349-359.
Romero, A., and Sánchez, M. (1995). Proc. Amer. Math. Soc. 123, n 9, 2831-2833.
Romero, A., and Sánchez, M. (1996). Bull. London Math. Soc. 28, n 5, 509-513.
Sánchez, M. (1997). Nonlinear Analysis TMA 30, 643-654.
Sánchez, M. (1997). Trans. Amer. Math. Soc. 349, n 3, 1063-1080.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aledo, J.A., Gálvez, J.A. On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes. General Relativity and Gravitation 35, 651–665 (2003). https://doi.org/10.1023/A:1022914101755
Issue Date:
DOI: https://doi.org/10.1023/A:1022914101755