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24 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
In the forthcoming sections we briefly review the main concepts yielding
to the definition of maximally symmetric totally geodesic spaces. To
start with, Sec. 2 is devoted to Killing vectors and their defining equations.
In Sec.3 we deal with the properties of maximally symmetric
spaces. Sec.4 is addressed to coordinate representations of maximally
symmetric spaces. The definition of maximally symmetric sub–spaces is
given in Sec. 5, with emphasis to the case of a hypersurface. In Sec.6
the characterization of totally geodesic hypersurfaces is given and the
explicit metric allowing for a maximally symmetric totally geodesic hypersurface
is reported. In Sec.7 maximally symmetric totally geodesic
sub–spaces are considered. In Sec.8 we established that the BTZ black
hole solution is not a locally maximally symmetric totally geodesic hypersurface
of the (3+1) conformally flat stationary axisymmetric PC
metric.
2. KILLING EQUATIONS AND
INTEGRABILITY CONDITIONS
Under a given coordinate transformation a metric is
said to be form–invariant if the transformed metric is the same
function of its argument as the original metric was of its
argument i.e. for all x. In general, at any given point
the law of transformation is thus,
for a form–invariant metric under transformation one replaces
by and consequently
which can be thought of as system of equations to determine as
function of Any transformation fulfilling (1) is called
an isometry. One way to look for solutions of (1) is to consider an
infinitesimal coordinate transformation
thus to the first order in Eq. (1) gives rise to the Killing equation
or, introducing covariant derivatives,