General analytic and infinitesimal deformations of immersions. I (Q1387281)

The author considers analytic and infinitesimal deformations of immersions of an \(n\)-dimensional smooth manifold in an \(m\)-dimensional, \(1 \leq n\leq m\), flat space with arbitrary signature, without any boundaries of the induced metric's variations and deformations of other sections of the vector bundle. Such deformations are called general analytic and infinitesimal deformations. The author shows that many (generalized) results of the theory of analytic and infinitesimal deformations of surfaces are correct in the case of general analytic and infinitesimal deformations. In particular, the following results are proved: theorem of existence and uniqueness of fields of rotates; analog of the fundamental theorem of the theory of surfaces for varied equations of Gauss, Peterson-Codazzi and Ricci; analogs of Allendörfer's theorem about the interdependency of the fundamental equations of the theory of surfaces for immersions of higher order.