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# 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

#### 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 *S*_{t}, with -δ < *t* < δ, given by an embedding

*R*_{t}*(u, v) = R(u, v)+ tU(u, v)*,

such that the first fundamental forms of *S*_{t} and *S* satisfy

*ds*^{2}_{t} = *ds*^{2} *+ O(t*^{2}).

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}.