Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

On maximally symmetric and totally geodesic spaces 27
so that and for which the first derivatives
at X takes all possible values, subject only to the antisymmetry
condition (4). In particular, in a one can choose a set of
Killing vectors such that
where upper indices in parenthesis denote the particular Killing vector
we are dealing with.
Moreover, for metric spaces that are isotropic about every point there
exist Killing vectors and that satisfy the
above initial conditions at X and at respectively. As a corollary
one has: W379, any space that is isotropic about every point is also
homogeneous.
Constancy of the scalar curvature.
Since at any given point, for maximally symmetric spaces, one can assign
to arbitrary values, the antisymmetric part of its coefficient
must vanish
Contracting and using the cyclic rule fulfilled by and the
definition of one obtains
The last equality has been added because of the antisymmetric character
in the first two indices and of the tensor From that part of
the above equation involving the Ricci tensor, contracting in and
one arrives at
Thus, substituting (17) in (16) one obtains the expression