On geometry of orbits of adapted projective frame space

Abstract

The current paper continues consideration of geometry of projective frame orbits started in the author’s article in the previous issue. The ndimensional projective space with a distinguished point (the center) is considered. The action of matrix affine group of order n on the adapted projective frame manifold is given. It is shown that the linear frames, i. e., bases of the tangent space, can be identified with the orbits of adapted projective frames under the action of some normal subgroup of this group. Two adapted frames are said to be equivalent if they belong to the same orbit. The strict perspectivity relation between two adapted frames is introduced. The proofs of the theorem on the Desargues hyperplane and of the criterion of equivalence are simplified. According to this criterion, two adapted frames in strict perspective are equivalent if and only if the Desargues hyperplane generated by these frames is passing through the center.