Geodesic
Type number and rigidity of fibred surfaces
Sbornik. Mathematics, Tome 192 (2001) no. 1, pp. 65-87

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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. 
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     author = {P. E. Markov},
     title = {Type number and rigidity of fibred surfaces},
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P. E. Markov. Type number and rigidity of fibred surfaces. Sbornik. Mathematics, Tome 192 (2001) no. 1, pp. 65-87. http://geodesic.mathdoc.fr/item/SM_2001_192_1_a3/