Methods for Finding Natural Operators

Abstract

We present certain general procedures useful for finding some equivariant maps and we clarify their application by solving concrete geometric problems. The equivariance with respect to the homotheties in GL(m) gives frequently a homogeneity condition. The homogeneous function theorem reads that under certain assumptions a globally defined smooth homogeneous function must be polynomial. In such a case the use of the invariant tensor theorem and the polarization technique can specify the form of the polynomial equivariant map up to such an extend, that all equivariant maps can then be determined by direct evaluation of the equivariance condition with respect to the kernel of the jet projection G ^r _m → G ¹ _m . We first deduce in such a way that all natural operators transforming linear connections into linear connections form a simple 3-parameter family. Then we strengthen a classical result by Palais, who deduced that all linear natural operators Λ^p T* → Λ^{p +1} T* are the constant multiples of the exterior derivative. We prove that for p > 0 even linearity follows from naturality. We underline, as a typical feature of our procedures, that in both cases we first have guaranteed by the results from chapter V that the natural operators in question have finite order. Then the homogeneous function theorem implies that the natural operators have zero order in the first case and first order in the second case. In section 26 we develop the smooth version of the tensor evaluation theorem. As the first application we determine all natural transformations TT* → T*T.The result implies that, unlike to the case of cotangent bundle, there is no natural symplectic structure on the tangent bundle.