Hypersurfaces in a Projective Curved Space

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This paper contains, with subsequent developments, the material presented in my lectures at Princeton University during the spring term of 1938. The reader finds here some account of the theory of hypersurfaces (both holonomic and non-holonomic) in a projective curved space. I have divided the paper into three chapters. In the first chapter (sections 1-6) the necessary notions and definitions are introduced and some objects defined which are needed in the next two chapters. In the second chapter (sections 7-13) I confine myself first to the non-holonomic case (sections 7-10) and consider the correlation associated with the given non-holonomic subspace. This correlation induces a projectivity which preserves, point by point, a projective direction (of order 2) of the hypersurface. The remaining sections of this chapter deal with the family of polarities, associated with the subspace (holonomic or non-holonomic) which in the holonomic case are at least of order 3 and enable us to define a canonical plane (section 12). A privileged polarity of this family is considered in the last section. In the third chapter (sections 14-18) the problem of normal direction and the connection induced by it are discussed. A normal direction (of order three in the holonomic general case) is found which, in the non-holonomic case, reduces to the normal direction mentioned above. (If the large space is flat and the hypersurface is holonomic then this normal direction is of course of order 4.) This normal direction induces a family of connections in the given hypersurface and enables us to construct an absolute connection. In the last section a projective Weyl's connection is introduced and discussed. Some of the problems considered here have already been treated by Schouten and Haantjes (Compositio Math. 3, pp. 1-51, 1936) in a completely different arrangement. These authors use the homogeneous coordinates in the curved space and consequently do not start with the usual definition of a projective transformation of an affine connection which gives rise to the notion of a projective curved space as used in dealing with the Princeton algorithm.