 
Summary: A SECOND ORDER SMOOTH VARIATIONAL PRINCIPLE ON
RIEMANNIAN MANIFOLDS
DANIEL AZAGRA AND ROBB FRY
Abstract. We establish a second order smooth variational principle valid
for functions defined on (possibly infinitedimensional) Riemannian manifolds
which are uniformly locally convex and have a strictly positive injectivity ra
dius and bounded sectional curvature.
1. Introduction and main result
It is well known that a continuous function defined on an infinitedimensional
manifold (or on a Banach space) does not generally attain a minimum in situations
in which there would typically exist minimizers if the function were defined on a
finitedimensional manifold (for instance when the infimum of the function in the
interior of a ball is strictly smaller than the infimum of the function on the boundary
of the ball). In fact, as shown in [1], the smooth functions with no critical points
are dense in the space of continuous functions on every Hilbert manifold (this result
may be viewed as a strong approximate version for infinite dimensional manifolds
of the MorseSard theorem). So, when we are given a smooth function on an
infinitedimensional Riemannian manifold we should not expect to be able to find
any critical point, whatever the overall shape of this function is, as there might be
none.
