Publications of the 1974

• A Generalized Hodge Theory

  Eiseman P.R., Stone A.P.

  Published in 1974

  For a given nonsingular vector one-form h with vanishing Nijenhuis tensor, there is an associated exterior derivative dh which satisfies a Poincare lemma and hence provides an h-dependent version of de Rham's theorem. The exterior derivative dh also has an adjoint δh with respect to the usual global inner product. This fact permits one to define a strongly elliptic self-adjoint second order differential operator Δh which is a generalization of the Laplace-Beltrami operator. Consequently one can then obtain a generalization of the classical Hodge decomposition theorem.

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• A Generalization of the Green's Operator on a Compact Manifold

  Eiseman P.R., Stone A.P.

  Published in 1974

  The Hodge theorem asserts the existence of a ρ-form λ such that Δλ = φ – H(φ), for an arbitrary C^∞ ρ-form φ. This equation has a unique solution orthogonal to all harmonic ρ-form, and the solution is denoted by G(φ). G is called the Green's operator. The purpose of this paper is to study generalized operators H_h and G_h where h is a vector l-form which is non singular and has vanishing Nijenhuis tensor [h,h].

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