A complete system of tensors of linear homogeneous second-order differential equations

Abstract

In an article appearing in the American Mathematical Monthlyt Lane called attention to the need in projective differential geometry of an absolute differential calculus for forms of higher degree than the quadratic. Such a calculus for forms of arbitrary degree had been published by the writer4 and it seemed that some adaptation to the theory of projective differential geometry would be desirable. However, it appeared more satisfactory to base a tensor theory directly upon the differential equations. Inasmuch as various systems of differential equations are used in the projective theory, a separate tensor theory will need to be constructed for each type. In this paper a study has been made of the differential equations upon which the ruled surface theory is based. The remarkable simplicity which has been introduced by tensor methods in the theory of Riemannian geometry and of algebraic invariants appears also in this theory. The Wilczynski theory, while general, has had the blemish of containing awkward unsymmetrical sets of equations involving cumbersome numerical constants. And with the improvement in notation there is an accompanying advantage in method, the simple formal tensor processes replacing the earlier ingenious methods. Some geometric interpretations of the tensors and invariants are given here in which the dependent variables yi(i = 1, * * * , n) are interpreted as nonhomogeneous co6rdinates of a point. This is in contrast to the convention peculiar to projective differential geometry of choosing a fundamental set of 2n independent solutions (y', * * , y), o =1, * * , 2n, and interpreting these as n points (y * * yn)), i 1, .. , n, in a homogeneous space of 2n-1 dimensions. An interpretation of our tensors in this latter space will constitute a generalization of Wilczynski's ruled surface theory.