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

Exact solutions in multidimensional gravity 125
where is an arbitrary metric with any signature
on the manifold and is a metric on
satisfying the equation
const, The functions are smooth. We
denote We put any manifold
to be oriented and connected. Then the volume
form and signature parameter
are correctly defined.
Let be a set of all non–empty subsets of The
number of elements in is For any
we denote
For fields of forms we consider the composite electromagnetic ansatz
with elementary forms of electric and magnetic types,
and For scalar functions we put
Thus and are functions on Here and below
The set consists of
elements where and Due
to (5) and (6): and for and
2.2. THE SIGMA MODEL
Let and (harmonic gauge).
It was proved in [6] that eqs. of motion for (0) and the Bianchi
identities: for fields from (3), (4)–(7), when some
restrictions are imposed, are equivalent to eqs. of motion for