Second order vectors and forms

Abstract

Given a manifold M, denote by E a C^∞-module of functions on M containing C^∞ and such that belonging to E is a local property (for instance all Borel, or locally bounded, or continuous, or C^p or C^∞ functions). Let L be a linear mapping from C^∞ to E (but not necessarily C^∞-linear), and denote by Γthe “squared field operator” associated to L:

$$\Gamma (f,g) = \frac{1}{2}[L(fg) - fLg - gLf].$$

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