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In the present paper, deformations of an immersion of an m-dimensional manifold M in Euclidean n-space Rn such that the Gauss image is fixed pointwise are studied. We call such deformations admissible deformations. The main theorem gives a necessary and sufficient condition for an immersed manifold to admit nontrivial admissible deformations. 1. Let M be an mr-dimensional C?? manifold and i: M x I -> a CO immersion where I is some interval, 0 E I. i(t) is defined by i(t)M = i(M x t) and we write i(t)M = M(t). M(t) is called a deformation of M(O). We assume the Gauss map F: M(t) -> G(m, n m) is regular for t E I. Let M(t) have the local expression x = x (y, . . . ; t) where x 1,... , x n are the rectangular coordinates in R n and y , ... , ym are local coordinates in M. We understand a(Da (D a= ai = aD ax(D = (y' (D.= a ayx' at h, i,j, . =1 ... ., n; K, , 1,...=1,it... M.m Suppose there exists a (1, 1)-tensor field a(y; t) satisfying aX = ax aox h. (1.2) Obviously the geometrical meaning of this equation is that the Gauss image of M(t) is fixed against t for each point of M. Let us call any such deformation M(t) an admissible deformation in this paper. As usual let ga be the components of the Riemannian metric of M(t) induced by the immersion, {, } be the Christoffel symbols, V be covariant differentiation with respect to the Christoffel symbols, B,uh = aXh/ayII and H,Ah be the second fundamental tensor. First we get from (1.2): a,aA.h { A, } a* = aV,,aGB + axGH,h. Received by the editors September 8, 1978. AMS (MOS) subject classifications (1970). Primary 53B25, 53C40.

Deformability of a submanifold in a Euclidean space whose image by the Gauss map is fixed