Variational Formulae

Variational formulae for geometric quantities are important in studying critical points of functionals on a manifold and flows of metrics. The fundamental question (similar to the question on existence of canonical metrics on a manifold) reads as: What are the optimal metrics (in a specified sense) on a smooth foliated manifold? Our goal here is to examine the actions on a manifold for different types of variations. Apart from varying among all metrics, we also deal with the case when the varying metric remains fixed along the distribution, and the complementary case when metric varies only along the distribution—preserving its orthogonal complement and the metric on it.