Nonrigidity of a class of two dimensional surfaces with positive curvature and planar points

Abstract

Existence of nontrivial infinitesimal bendings is established for an orientable surface with boundary S ⊂ R 3 S\subset \mathbb {R}^3 that has positive curvature except at finitely many planar points and such that H 1 ( S ) = 0 H_1(S)=0 . As an application, we show that any neighborhood of such a surface S S (for the C k C^k topology) contains isometric surfaces that are noncongruent.