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

On maximally symmetric and totally geodesic spaces 29
4. REPRESENTATION OF MAXIMALLY
SYMMETRIC SPACES
In general, one may pose the question if there exists a system of
coordinates in a space of constant curvature such that the metric is of
the form
where C is a diagonal matrix with non–zero diagonal elements
for depending upon the signature.
The substitution in the Riemann tensor yields
in the right hand side there is no summation neither in nor in
From the first equations one arrives at
where depend only on their arguments. Substituting in the
second equations one obtains
where is no summation in The constants and are arbitrary,
and the constant curvature K occurs to be related with them through
Hence, one can give various alternative representations of the metric
of the studied space. In particular, the Riemman form arises when
is different from zero; accomplishing first a shifting of the
followed by scaling transformations
one arrives at the Riemann form
Representation as embedding in a
Let us consider the metric (26) given as