Abstract
An analytic method, which Wu called the “Bochner technique,” has been used for fifty years to describe global Riemannian and Kähler geometries. We use this method to describe conformally Killing vector fields and harmonic timelike vector fields on a Lorentzian manifold and to study hydrodynamic models of the Universe, the existence of closed spacelike sections, and the possibility of fibering Lorentzian manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Wu, “The Bochner technique,” in:Proc. Beijing Symp. Differential Geometry and Differential Equations (Aug. 18–Sept. 21, 1980) (S. S. Chern and Wen-tsuen Wu, eds.), Gordon and Breach, New York (1982), pp. 929–1071.
P. H. Berard,Bull. Am. Math. Soc.,19, 371–406 (1988).
W. Matthias,Bonn. Math. Schr., No. 198, 1–58 (1989).
I. J. Muzinich,J. Math. Phys.,27, 1393–1397 (1986).
M. Blau,Rep. Math. Phys.,25, 109–116 (1998).
A. Romero and M. Sanchez,Bull. Am. Math. Soc.,123, 2831–2833 (1995).
A. Romero and M. Sanchez,Pacific J. Math.,186, 141–148 (1998).
S. E. Stepanov,Izv. Vyssh. Uchebn. Zaved. Fizika, No 6, 82–86 (1993).
S. E. Stepanov,St. Petersburg Math. J.,10, 703–714 (1998).
S. E. Stepanov,Diff. Geom. Mnogoobr. Figur,27, 107–111 (1996).
S. E. Stepanov,Tensor,58, (n.s.), 245–255 (1997).
J. Beem and P. Ehrlich,Global Lorentzian Geometry, Marcel Dekker, New York (1981).
A. Besse,Manifolds All of Whose Geodesics Are Closed, Springer, Berlin (1978).
C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation, Freeman, San Francisco (1973).
D. Kramer, H. Stephani, E. Herlt, and M. MacCallum,Exact Solutions of Einstein's Field Equations (E. Schmutzer, ed.), Cambridge Univ. Press (1980).
J. L. Synge,Relativity: The General Theory, North-Holland, Amsterdam (1960).
S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time, Cambridge Uni. Press, Cambridge (1973).
S. E. Stepanov,Theor. Math. Phys.,111, 419–427 (1997).
S. E. Stepanov,Izv. Vyssh. Uchebn. Zaved. Fizika, No. 5, 90–93 (1996).
B. L. Reinhart,Differential Geometry of Foliations, Springer, Berlin (1983).
S. E. Stepanov,Izv. Vyssh. Uchebn. Zaved. Matem., No. 1, 61–68 (1999).
Sh. Kobayashi and K. Nomizu,Foundations of Differential Geometry, Vol. 1, Wiley, New York (1963).
L. Eisenhart,Riemann Geometry, Princeton Univ. Press, Princeton (1926).
V. D. Zakharov,Gravitational Waves in Einstein's Theory [in Russian], Nauka, Moscow (1972); English transl., Halsted (Wiley), New York (1973).
S. E. Stepanov,Russ. Math.,40, No. 9, 50–55 (1996).
Helio V. Fagundes,Gen. Relat. Gravit.,24, 199–217 (1992).
R. Penrose,Structure of Space-Time, Benjamin, New York (1968).
Author information
Authors and Affiliations
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 482–496, March, 2000.
Rights and permissions
About this article
Cite this article
Stepanov, S.E. An analytic method in general relativity. Theor Math Phys 122, 402–414 (2000). https://doi.org/10.1007/BF02551253
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02551253