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

138 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where the 4–dimensional connection satisfies and
is the zero–form
Next we calculate the components of the curvature 2–form obtaining
Here is the 4–dimensional curvature and
is the covariant derivative of the zero–form in the 4–dimensional
manifold, which is given by
Now we calculate the contribution to the action (10) which is
quadratic in Separating the and 5 parts, this term splits into
two pieces, namely where
we subsequently substitute the equations (23), (24) for the Riemann 2–
forms. The second expression involves the contribution of two covariant
derivatives which are transformed into surface terms plus non–derivative
terms using an integration by parts together with the identity
We obtain