The projective differential geometry of systems of linear homogeneous differential equations of the first order

Abstract

The tedious calculations that occasionally occur in studying projective differential geometry by means of the invariants and covariants of a completely integrable system of linear homogeneous differential equations are admittedly an unpleasant feature of this method. But it seems that labor can sometimes be avoided by reducing the system of equations to a system of the first order. Such a system proves to be especially suitable for the study of a configuration composed of a set of varieties generated by linear spaces, with the generators in correspondence. The present paper is concerned for the most part with the projective differential geometry that can be studied by means of a system of n+1 linear homogeneous differential equations of the first order in n + 1 dependent variables and one independent variable. The transformation of dependent variables that is used is determined by the configuration to be studied, which in any case consists of a set of varieties each of which is generated by oo 1 linear spaces, with the generators in correspondence. The precise number of possible configurations of this type in a space of a given number of dimensions is determined, but by imposing the condition that some of the varieties shall be covariant to the rest of them the range of applicability of the method can be widely extended. Certain curves called intersector curves on a variety are defined, and the locus of their tangents is investigated. A section is devoted to geometry in a space S5 of five dimensions. The intersection of a configuration with a hyperquadric in S5 yields a geometric interpretation in ordinary ruled space. Another section is taken up with a pair of ruled surfaces in ordinary space, with their generators in correspondence. A complete system of invariants is furnished therefor, and a new canonical form of the differential equations is established. Finally a system of k(n+ 1) linear homogeneous partial differential equations of the first order in n+1 dependent variables and k independent var-

Locations

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