Euclidean conformally flat submanifolds in codimension two obtained as intersections

Abstract

We characterize a large family of codimension two euclidean conformally flat submanifolds in the class of isometrically rigid ones and construct explicit examples. In [DF] we showed that generic conformally flat submanifolds with codimension two in euclidean space, f: M -* R+2, n > 5, can be divided into three classes, namely, the surface-like ones, those which admit locally a continuous 1-parameter family of isometric deformations, and those which are locally isometrically rigid. In addition, we generated explicit examples of elements in the first and second classes. On the other hand, no example came out from our rather elaborate descriptions of submanifolds in the third class, thus naturally raising the question of whether such submanifolds really exist. Our main achievement here is give a positive answer to this question. In what follows we make free use of definitions and results in [DF]. Our approach is the following. First we give a parametric description of a large family of generic conformally flat submanifolds, each of which is completely determined by two curves and two functions in one variable. Then, we prove that almost all elements in the family are locally isometrically rigid. Next, we give a brief indication of how they can be obtained by a geometric procedure, namely, as intersections starting from two flat hypersurfaces. We conclude this note by showing that a particular selection of curves and functions yields explicit examples. Consider a pair of smooth curves aj: Ij C R -, Rn+2, 1 1 everywhere. Now take V2 = I, x 12 small enough so that (1) 0(u,v) = 0(u,O) + jOv (us)ds, 0(u,O) = -I|a (u) 1, is positive, Ov being the unique (and necessarily smooth) solution of the integral equation of Volterra type Ov(u, ) =(u), a2(V)) 0(U, O) fa (2(v), a2(t))dt (2) J O j (j; (al1 (v), a2(t)) dt) Ov (u, s)ds. Received by the editors January 26, 1996 and, in revised form, April 30, 1997. 1991 Mathematics Subject Classification. Primary 53B25; Secondary 53C25.