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

On maximally symmetric and totally geodesic spaces 35
7. TOTALLY GEODESIC SUB–SPACES
A immersed in a is called totally geodesic subspace
of if all the geodesics of are geodesics of
A necessary and sufficient condition for a immersed in a
be totally geodesic is that
where see (54), are given by
where denote the components of a particular choice
of mutually orthogonal one to another and normal to vectors. The
functions and are invariants for transformations of coordinates
in are the components of a symmetric tensor in the
and are components of a vector. Hence it follows from (69), that for
each value of the are the components of a symmetric tensor in
One can write
In E186 one encounters the following statement: the sub–spaces
const., are totally geodesic in if the metric of
can be decomposed as
where the functions are independent of
In general, are functions of and
7.1. MAXIMALLY SYMMETRIC TOTALLY
GEODESIC SUB–SPACES
If metric components of the above metric are independent of
that is the sub–spaces are maximally symmetric
totally geodesic in if the metric of can be decomposed
as