Variational Formulae

Abstract

We study extrinsic geometry (properties depending on the second fundamental form) of a codimension-one foliation subject to ($\mathcal{F}$-truncated) variations of metrics along the leaves. In Sect. 2.3.1 we develop formulae for the deformation of geometric quantities as the Riemannian metric varies along the leaves. Then, in Sect. 2.3.2, we study variation properties of the functionals depending on the principal curvatures of the leaves and the $\mathcal{F}$-truncated families of metrics, in particular, for conformal metrics along the leaves. The last section of Chap. 2 contains applications to umbilical foliations and minimization of the total bending of the unit normal vector field.