VARIATIONAL PROBLEMS FOR INTEGRAL INVARIANTS OF THE SECOND FUNDAMENTAL FORM OF A MAP BETWEEN PSEUDO-RIEMANNIAN MANIFOLDS

Abstract

We study variational problems for integral invariants, which are defined as integrations of invariant functions of the second fundamental form, of a smooth map between pseudo-Riemannian manifolds. We derive the first variational formulae for integral invariants defined from invariant homogeneous polynomials of degree two. Among these integral invariants, we show that the Euler–Lagrange equation of the Chern–Federer energy functional is reduced to a second order PDE. Then we give some examples of Chern–Federer submanifolds in Riemannian space forms