Communications in Analysis and Geometry
Volume 11 (2003)
Infinitesimal Bendings of Homogeneous Surfaces with Nonnegative Curvature
Pages: 697 – 719
This paper deals with infinitesimal bendings of a surface S in a neighborhood of a point p ∈ S. More precisely, consider a surface S embedded in 3 and given by parametric equation
R(u, v) = (x(u, v), y(u, v), z(u, v)) ∈ 3,
with (u, v) ∈ 2 and p = R(0, 0) = 0. An infinitesimal bending of S is a deformation St, with -δ < t < δ, given by an embedding
Rt(u, v) = R(u, v)+ tU(u, v),
such that the first fundamental forms of St and S satisfy
ds2t = ds2 + O(t2).
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 3.