  Solving field equations in the context of higher curvature gravity theories
is a formidable task. However in many situations, e.g., in the context of
$f(R)$ theories the higher curvature gravity action can be written as
Einstein-Hilbert action plus a scalar field action. We show that not only the
action but the field equations derived from the action are also equivalent
provided the spacetime is regular. We also demonstrate that such equivalence
continues to hold even when gravitational field equations are projected on a
lower dimensional hypersurface. We have further depicted explicit examples in
which the solutions for Einstein-Hilbert and a scalar field system lead to
solutions of the equivalent higher curvature theory. The same, but on the lower
dimensional hypersurface, has been illustrated in the reverse order as well. We
conclude with a brief discussion on this technique of solving higher curvature
field equations.
