A note on local minimizers of energy on complete manifolds

In this paper, we study the geometric rigidity of complete Riemannian manifolds admitting local minimizers of energy functionals.
More precisely, assuming the existence of a non-trivial local minimizer and under suitable assumptions, a Riemannian manifold under consideration must be
a product manifold furnished with a warped metric.
Secondly, under similar hypotheses, we deduce a geometrical splitting in
the same fashion as in the Cheeger-Gromoll splitting theorem and we also get information about local minimizers.