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

126 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where
is the target space metric,
Here and and
3. TODA–TYPE SOLUTIONS FROM
NULL–GEODESIC METHOD
It is known that geodesics of the target space equipped with some
harmonic function on a three–dimensional space generate a solution to
the equations [2]. Here we apply this null–geodesic method to
our sigma-model and obtain a new class of solutions in multidimensional
gravity. Action (8) may be also written in the form
where
on is defined as follows
Consider exact solutions to field equations for (11). We put
where is a smooth function, is a
harmonic function on satisfying
for all
Let all factor spaces are Ricci-flat and cosmological constant is zero.
Then, the potential is zero and the field equations corresponding to (11)
are satisfied identically if obey the Lagrange equations for