Infinitesimal higher order bendings of multidimensional surfaces in spaces of constant curvature

Infinitesimal bendings of order $r\geqslant1$ are considered, including analytic bendings ($r=\infty$), of an $n$-dimensional surface $F$ in an $m$-dimensional ($1\leqslant n<m$) space $W$ of constant curvature. It is proved that to any solution of an $r$ times formally varied system of Gauss–Codazzi–Ricci equations there corresponds an infinitesimal bending of order $r$ of the surface $F$ in $W$. A general form is established for solutions of this system that determine infinitesimal motions of various orders. By using these results we obtain criteria for rigidity and nonrigidity of order $r\leqslant1$, and also for analytic bendability and nonbendability of a class of multidimensional surfaces of codimension $p\geqslant1$ in flat spaces, which contains, in particular, Riemannian products of hypersurfaces.
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