Tensor Fields with Finite Bases

Abstract

We will establish the general principle (with applications) that any module of tensors or tensor densities of any mixed type on a compact manifold has a finite basis whenever the tensors are complex-holomorphic or, more generally, in the real case, are in a certain general manner. Our concept of harmonic is rather more general than the one current in the theory of cohomology and no such restrictions as skew-symmetry are required for us. We will also operate, even more generally, on the universal covering space of the given one, and the finite bases will also occur for finite groupings of tensors on the covering space which reproduce themselves by arbitrary bounded representations of the fundamental group. Our consequences will be mainly analytical, but Theorem 8 also suggests the purely topological result of associating Betti numbers {Br} with every bounded irreducible representation of the fundamental group, the ordinary Betti numbers being those corresponding to the identity representation. This ought to apply to general complexes, and further statements bearing on this question will be made in a subsequent paper. 1. Holomorphic tensors THEOREM 1. (i) On a compact complex manifold V = V2k the module of all holomorphic tensor densities (~~~~~~~~~~~~~~~~~~~~~~C 1 ) tp l; s a