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26 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
then the function if it exits, can be constructed as a Taylor series
in within a finite neighborhood of X, thus any Killing vector
of the metric can be expressed as
where and the same for all Killing vectors at the
neighborhood of X, are functions of the metric and X. Nevertheless,
whether (4) is solvable for a given set of initial data and
depends on the above integrability conditions (10), imposing
linear relations among and at any given point X. The maximal
dimension of the space of initial data constructed on
being is reduced by the number of relations
(10). Thus, if there were no relations (10) on the data and
then each Killing vector of a given was uniquely
determined by the independent quantities and in number
equal respectively to and Hence, a allows for at most
an parameter group of Killing vectors
The so-called maximally symmetric space or space of constant
curvature, admits the maximal number of Killing vectors.
Hence, the equations (10) do not impose conditions on and
at any point X, and therefore they acquire respectively and
arbitrary values at X. The vanishing of the factor multiplying
in (10) yields to the constancy of the scalar curvature R.
3. MAXIMALLY SYMMETRIC SPACES
A maximally symmetric space is, by definition, a space which admits
a continuous group of motions of parameters. On the other
hand, a maximally symmetric space is characterized, according to Weinberg,
by the following property:W379, a maximally symmetric space
is necessarily homogeneous and isotropic about all points. Weinberg
worked out in details the structure of the Killing vectors, which arises
from the the homogeneous and isotropic properties of a considered space:
W378:A metric space is said to be homogeneous if there exit infinitesimal
isometries (2) that carry any point into any other point
in its immediate neighborhood. That is, the metric must admit
Killing vectors that at any point take all possible values. In particular,
in a one can choose a set of Killing vectors
with
W378: A metric space is said to be isotropic about a given point if
there exit infinitesimal isometries (2) that leave the point X fixed,