  We derive the condition on f(R) gravities that admit Killing spinor equations
and construct explicit such examples. The Killing spinor equations can be used
to reduce the fourth-order differential equations of motion to the first order
for both the domain wall and FLRW cosmological solutions. We obtain exact "BPS"
domain walls that describe the smooth Randall-Sundrum II, AdS wormholes and the
RG flow from IR to UV. We also obtain exact smooth cosmological solutions that
describe the evolution from an inflationary starting point with a larger
cosmological constant to an ever-expanding universe with a smaller cosmological
constant. In addition, We find exact smooth solutions of pre-big bang models,
bouncing or crunching universes. An important feature is that the scalar
curvature R of all these metrics is varying rather than a constant. Another
intriguing feature is that there are two different f(R) gravities that give
rise to the same "BPS" solution. We also study linearized f(R) gravities in
(A)dS vacua.
