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

28 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
The constancy of R results from the Bianchi identity
Contracting in and one obtains
contracting again in and one gets
substituting (17) in the above equation, one arrives at
Hence any space of three or more dimensions in which (17) holds everywhere
will have R constant. Moreover, for maximally symmetric spaces
of two dimensions the scalar curvature R occurs to be also constant. A
necessary and sufficient condition that the curvature at every point of
space be independent of the orientation is that the Riemann tensor is
expressible as (18).
Introducing the curvature constant K, one has
On the other hand, via the homogeneous property at a given point
in a maximally symmetric space, there exist independent Killing vectors
for which takes any values one likes, thus
which by cyclic permutation of and gives
which, by virtue of the properties of the Riemann tensor, is identically
zero, and therefore it gives no information.
It becomes then clear that maximally symmetric spaces are unique in
the sense that any two maximally symmetric spaces of the same signature
and the same constant curvature are isometric and have the same
intrinsic properties, and that the equations of the isometric correspondence
involve arbitrary constants.