Differential Forms in Manifolds with Boundary

Abstract

The theory of harmonic forms in Riemannian manifolds may be regarded as a generalization of potential theory. It is therefore natural that the boundary value problems of this theory which generalize the classical Dirichlet and Neumann problems should play an important role in the theory. Such theorems were conjectured by Tucker (4a). The present paper is concerned with some theorems regarding closed forms, which are necessary for the proof of the boundary value existence theorems for harmonic fields in a Riemannian manifold with boundary. In an accompanying paper, by D. C. Spencer and the author, the results of this paper will be used in the proofs of these theorems. We consider closed forms in a finite regular manifold with boundary, and prove results which generalize two theorems of G. de Rham (3) for a closed manifold. To extend the technique of de Rham, we make use of the relative, or modular, homology theory introduced by Lefschetz (2). The main results of the paper are contained in Theorems 2 and 6. Theorems 1, 3, and 5 are analogues of de Rham's first theorem, and Theorems 2, 4 and 6 of his second theorem. Theorems 3 to 6 inclusive were conjectured in substantially their present form by Tucker (4b).