Tensors associated with a pair of quadratic differential forms

Abstract

A pair of quadratic differential forms in n dimensions determine at least partly at each point of space, a system of n invariants and a set of n orthogonal unit vectors, with a one to one correspondence between the two sets. It is found very convenient to use these invariants and vectors as auxiliary quantities in terms of which tensors associated with the two forms can be expressed. In order that a tensor, which is a rational function 1 1 A tensor that is a rational function of a set of quantities is defined as a tensor whose components are rational functions of this set of quantities, the functional relationship being invariant under coördinate transformations. of these auxiliary quantities and their higher derivatives, be also a rational function of the components of the original two forms and their higher derivatives, it is necessary and sufficient that this tensor be invariant under a certain group of transformations on the auxiliary quantities. The Galois theory of groups is applied and a generalization to tensors is obtained of the theorem that every rational symmetric function of the roots of an algebraic equation is a rational function of the coefficients of the equation. The tensors that are rational functions of the components of the original two forms are considered in detail. Certain systems of fundamental tensors, symmetric relative to the auxiliary quantities, are introduced, such that every rational tensor function of the auxiliary quantities and symmetric relative to these quantities, can be expressed uniquely in terms of these fundamental tensors. Only tensors of even order are rational functions of the two forms. If one form is invariantively related to the other, the theory here developed becomes a theory for a single form.