Differentiable mapping of affine Q[m] and projective P[n] spaces (m>n)

Abstract

The urgency of work is caused by necessity of detailed studying of differentiable mappings of multivariate space. The main aim of the research is to study differentiable mappings of V[m, n] of affine space Q[m] to projective space P[n] (m>n); to consider mapping not only by analytical methods but also geometrically with the help of the attached geometrical images. Methods of research. The basic method of research is the method of external forms Cartan in local differential geometry and G. F. Lapteva's theoretical-group method. These methods assume local studying of the considered objects and use of functions of a class C[∞]. Results. The authors have obtained the differential equations of internal fundamental geometrical objects of the first and the second orders of differentiable mappings of space Q[m] in manifolds singular and nonsingular null-pairs space P[n]. The invariant geometrical images were found analytically and geometrically. The images were determined by the fundamental object components which helped in solving the problem of invariant determining the Q[m] space mapping in manifolds of null-pairs of P[n] space.

Keywords

• differentiated mapping
• multidimensional affine and projective spaces