Genuine deformations and submanifolds (Q1769133)

The authors introduce the concept of genuine isometric deformation of a Euclidean submanifold and they describe the geometric structure of submanifolds that admit deformations of this kind. Definition: An isometric immersion \(\widehat{f}:M^n\mapsto\mathbb R^{n+q}\) is a genuine deformation of a given isometric immersion \(f:M^n\mapsto\mathbb R^{n+p}\) if there is no open subset \(U\subset M^n\) along which the restictions \(f_{| U}\) and \(\widehat{f}_{| U}\) extend isometrically. The authors prove that any pair of submanifolds in low codimension determined by a genuine transformation is mutually ruled (with the same rullings) and they give sharp estimate for the dimension of the rullings. In addition they show that the relation discussed in the sequel between the normal bundles and second fundamental forms that exists for any pair of mutually real submanifolds must satisfy strong additional conditions. The main result of the paper is as follows. Theorem: Let \(\widehat{f}:M^n\mapsto\mathbb R^{n+q}\) be a genuine deformation of \(f:M^n\mapsto\mathbb R^{n+p}\) with \(p+q<n\) and \(\min\{p,q\}\leq 6\). Then \(f\) and \(\widehat{f}\) are mutually \(D^d\)-ruled along each connected component of an open dense subset of \(M^n\) with \(d\geq n-p-q+3l\) unless \(\min\{p,q\}=6\) and \(l=0\) in which case \(d=n-p-q-1\) and the bundle isometry \(T:L^l\mapsto L^l\) satisfies two conditions: (\(C_1\)) The isometry \(T\) is parallel and preserves second fundamental forms (\(C_2\)). The subbundles \(L\) and \(\widehat{L}\) are parallel along \(D\) in the normal connections.