The Differential Invariants of a Two-index Tensor

Riemannian geometry, based upon a metric form ds = gijdxdx', gives us the curvature tensor R)u as the sole basic differential invariant of the space, and of the symmetric tensor gy. The general tensor g^ can be broken up into the sum of two irreducible components, namely the symmetric and antisymmetric portions defined respectively by 2giij)=gij+gji and 2g[ij]=gij--gji. The latter disappears in constructing ds; but the general differential invariants of gij must necessarily be composed of those derivable from g(ij) (the curvature tensor above), from gun, and a group of mixed invariants dependent upon both. I t is proposed to investigate the general problem by use of a well known and easily proved fundamental lemma of the calculus of variations : The Euler equations derived from a variational principle are tensor-invariant under the group of transformations which leaves the original integral invariant. Actually the equations as directly obtained state that a certain covariant vector vanishes. Given the tensor ga{x • • • x) we first introduce two (implicit) absolute parameters u, v, and construct the variational problem