ON SOME L1-FINITE TYPE (HYPER)SURFACES IN Rn+1

We say that an isometric immersed hypersurface x : M-n -> Rn+1 is of L-k-finite type (L-k-f.t.) if x = Sigma(p)(i=0) x(i) for some positive integer p < infinity, x(i) : M -> Rn+1 is smooth and L(k)x(i) = lambda(i)x(i), lambda(i) is an element of R, 0 <= i <= p, L(k)f = trP(k) o del(2)f for f E C-infinity(M), where P-k, is the kth Newton transformation, del(2)f is the Hessian of f, LkX = (L(k)x(1), . . . , L(k)x(n+1)), x = (x(1), . . . , x(n+1)). In this article we study the following (hyper)surfaces in Rn+1 from the view point of L-1-finiteness type: totally umbilic ones, generalized cylinders S-m(r) x Rn-m, ruled surfaces in Rn+1 and some revolution surfaces in R-3.