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LOCAL RIGIDITY OF SURFACES IN SPACE FORMS MICHAEL T. ANDERSON
 

Summary: LOCAL RIGIDITY OF SURFACES IN SPACE FORMS
MICHAEL T. ANDERSON
Abstract. We prove that isometric embeddings of closed, embedded surfaces in R 3 are locally
rigid, i.e. they admit no non-trivial local isometric deformations, answering a question in classical
di erential geometry. The same result holds for abstract surfaces embedded in constant curvature
3-manifolds, provided a mild condition on the fundamental group holds.
1. Introduction.
A well-known question in classical di erential geometry is whether isometric embeddings of closed
surfaces  in Euclidean space R 3 are locally rigid. Thus, suppose is a smooth Riemannian metric
on  and
F : (; ) ,! (R 3 ; g 0 )
is a smooth isometric embedding, where g 0 is the Euclidean metric. If F s is a curve of isometric
embeddings of  in R 3 with F 0 = F , does F s necessarily arise from a curve  s in the group of
isometries of R 3 , i.e. F s =  s F 0 ? Of course open surfaces such as a at plane, are not locally
rigid. Thus, throughout the paper, we assume that  is compact and connected; we also asume
that the curve F s is C 1 in s.
It is useful to rst describe some background setting and results. In case  is strictly convex, or
equivalently when the Gauss curvature K of is positive, it is well known that (; ) is locally
rigid. In fact surfaces of strictly positive curvature are globally rigid, in that isometric embeddings
are uniquely determined up to rigid motion of R 3 . Classical proofs of this fact have been given by

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics