Communications in Analysis and Geometry

Volume 11 (2003)

Number 4

Infinitesimal Bendings of Homogeneous Surfaces with Nonnegative Curvature

Pages: 697 – 719

DOI: http://dx.doi.org/10.4310/CAG.2003.v11.n4.a3

Author

Abdelhamid Meziani

Abstract

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.

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