The purpose of this paper is to propose the implementation of some methods
from algebraic geometry in the theory of gravitation, and more especially in
the variational formalism. It has been assumed that the metric tensor depends
on two vector fields, defined on a manifold, and also that the gravitational
Lagrangian depends on the metric tensor and its first and second differentials
(instead on the partial or covariant derivatives, as usually assumed).
Assuming also different operators of variation and differentiation, it has
been shown that the first variation of the gravitational Lagrangian can be
represented as a third-rank polynomial in respect to the variables, defined in
terms of the variated or differentiated vector fields. Therefore, the solution
of the variational problem is found to be equivalent to finding all the
variables - elements of an algebraic variety, which satisfy the algebraic
equation.Comment: Latex (amsmath style), 10 pages, no figures; to appear in Proceedings
of the Fifth International Workshop on Complex Structures and Vector Fields,
(September 2000, St.Konstantine,Bulgaria), World Scientific,Singapore, 2001,
eds. K.Sekigawa, S.Dimie
