Infinitesimally Bonnet Bendable Hypersurfaces

The classical Bonnet problem is to classify all immersions f: M^2→ℝ^3 into Euclidean three-space that are not determined, up to a rigid motion, by their induced metric and mean curvature function. The natural extension of Bonnet problem for Euclidean hypersurfaces of dimension n≥ 3 was studied by Kokubu (Tôhoku Math J 44:433–442, 1992). In this article, we investigate an infinitesimal version of Bonnet problem for hypersurfaces with dimension n≥ 3 of any space form, namely, we classify the hypersurfaces f:M^n→ℚ_c^n+1 , n≥ 3 , of any space form ℚ_c^n+1 of constant curvature c , for which there exists a (non-trivial) one-parameter family of immersions f_t:M^n→ℚ_c^n+1 , with f_0=f , whose induced metrics g_t and mean curvature functions H_t coincide “up to the first order," that is, ∂ /∂ t|_t=0g_t=0=∂ /∂ t|_t=0H_t.