Remarks on natural differential operators with tensor fields

Abstract

We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.

Remark on bilinear operations on tensor fields

“…In particular, we have verified that [3] has fully classified all bilinear operators listed in (1) on manifolds of dimension at least two commuting with Lie derivatives, i.e. satisfying the algebraic property (2), without any further continuity or locality assumptions.…”

“…In the rest of this short note, we prove the following theorem: Theorem 1. All bilinear operators of the seven types listed in (1) defined on compactly supported sections and commuting with Lie derivatives are bilinear natural differential operators classified in [3]. In particular, no continuity or locality have to be assumed.…”

“…This is a truly remarkable theorem. Let us look at the example of a functional as in (3). First, the theorem says that such a functional has to be defined by an integral operator with a kernel.…”