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The main question is whether a given surface S admits nontrivial infinitesimal bendings in a neighborhood of p. By nontrivial infinitesimal bendings we mean those bendings that are not induced by the rigid motions of the ambient space R3. The study of this question and related problems is very old and goes back at least to Darboux ([D]). Local and global aspects of such problems, as well as the physical applications (elasticity of thin shells for example), have been studied in [A], [BG], [C], [D], [E1], [E2], [EU], [K], [P], [S1], [S2], [S3], [S], [U1], [U2], [V]. Let K(u, v) be the gaussian curvature of S. WhenK(0) = 0, the infinitesimal bendings of S are well understood (see [D] or [V]). The case K(0) = 0 leads to equations with singularities. In 1948, Efimov [E3] proved that the surface (u, v, u9 +λu7v2 + v9), with λ ∈ R a transcendental number, is rigid under real analytic bendings. Later on, he proved that most real analytic surfaces are rigid under real analytic bendings.

Infinitesimal Bendings of Homogeneous Surfaces with Nonnegative Curvature