On an integral formula for differential forms and its applications on manifolds with boundary

摘要

Given two k -forms α and β on a compact Riemannian manifold M with boundary ∂M, we derive an identity relating ∫_M(+<δα;δβ>-<▽α;▽β>)- ∫_M to an integral on the boundary ∂M. Herein F_k is a bundle endomorphism depending only on the Riemannian curvature tensor. Essential is how the tangential and normal parts of α and β, respectively their derivatives, appear in the integrand of the boundary integral. The identity gives a very simple proof for many classical results, which require that α = β and also that the tangential part αT (or the normal part αN) vanishes on ∂M. Moreover we generalize some of the boundary conditions in Gaffney inequality and in nonexistence theorems for harmonic fields.