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

On maximally symmetric and totally geodesic spaces 33
and
where the immersion of is defined through the functions
with the Jacobian matrix of rank
Let Greek indices take the values while indices
run and run In one can chose a system
of real unit vectors mutually orthogonal to one another,
and normal to
where Differentiating covariantly these equations with respect
to the metric of i.e. with respect to and the metric
one arrives at
which can be rewritten as
where is the Christoffel symbol with respect to and evaluated
at points of
Specializing the theory for a space of and its
hypersurface the metric of the space is given by
while the metric of the hypersurface is given as
The immersed hypersurface is denned by equations of the form
A unit normal to vector has to fulfill the orthogonality condition