Type number and rigidity of fibred surfaces

Infinitesimal $l$-th order bendings, $1\leqslant l\leqslant\infty$, of higher-dimensional surfaces are considered in higher-dimensional flat spaces (for $l=\infty$ an infinitesimal bending is assumed to be an analytic bending). In terms of the Allendoerfer type number, criteria are established for the $(r,l)$-rigidity (in the terminology of Sabitov) of such surfaces. In particular, an $(r,l)$-infinitesimal analogue is proved of the classical theorem of Allendoerfer on the unbendability of surfaces with type number $\geqslant 3$ and the class of $(r,l)$-rigid fibred surfaces is distinguished.