On Extensions of Infinitesimal Deformations

The classical Allendoerfer’s local rigidity result assures that any isometric immersion f : M → Q c with type number ρf ≥ 3 everywhere is isometrically rigid. Here and throughout the paper, M stands for a connected n−dimensional Riemannian manifold and Qc denotes a complete simply connected Riemannian manifold of constant sectional curvature c. On the other hand, Dajczer and Rodŕıguez ([1]) have shown that the same type number condition also guarantees infinitesimal rigidity. Thus, “generically speaking”, isometric rigidity and infinitesimal rigidity are the same property in low codimension. A stronger concept of isometric rigidity was considered by Dajczer and Tojeiro in [2]. Given an isometric embedding f : M → Q c and an isometric immersion g : M → Q c , they proved, under some weak regularity conditions on g, that g must be a composition g = h ◦ f , with ρf ≥ p + 2 everywhere. Here h : U ⊂ Q c → Q c is an isometric immersion and U an open subset containing f(M). In view of [1], it is thus natural to look for an infinitesimal version of this stronger isometric rigidity result. The main purpose of this paper is to show that, in this context, any infinitesimal deformation of the composition g must be the restriction to f(M) of an infinitesimal deformation of the extension h, when we restrict ourselves to the case of codimension p = 2. To state our main result, we first need some definitions. We say that an infinitesimal deformation Z of a composition g = h◦f admits an extension along h, if Z = Z ◦f , where Z is an infinitesimal deformation of h|U ′ , for some open subset f(M) ⊂ U ′ ⊂ U . The isometric immersion g is said to be 1-regular if the normal space spanned by its second fundamental form has constant dimension.