A projective generalization of metrically defined associate surfaces

Abstract

In the metric differential geometry of surfaces in ordinary space, two surfaces are said by Bianchi to be associatet if the tangent planes at corresponding points are parallel and if the asymptotic curves on either surface correspond to a conjugate net on the other. It is the purpose of this paper to develop a projective generalization of the relation of associateness of surfaces. Since associate surfaces are parallel in the metric sense, it will first be necessary to provide a projectively defined substitute for the property of metric parallelism. We shall employ as the basis of our study in this paper a projective generalization of euclidean parallelism of surfaces which the author has developed in his Chicago doctoral dissertation. In ?2, after stating a definition of projective parallelism of surfaces and briefly explaining this idea, we introduce a canonical form of our system of differential equations employed in the study of projectively parallel surfaces in ordinary space. In ?3 we formulate a definition of projectively associate surfaces and investigate to some extent their properties and relations. A more general type of associateness which may be conveniently termed modified projective associateness is introduced in ?4, and a somewhat different canonical form of our system of differential equations is employed in its study. Finally, in ?5, we consider a rather general completely integrable system of partial differential equations, namely, the system for two surfaces in the general analytic one-to-one point correspondence in ordinary projective space S3, and a group of transformations that leaves this configuration invariant. We then reduce this system of equations to a new canonical form, and employ it to continue briefly the study of modified projective associateness introduced in the preceding section.