Product submanifolds with pointwise 3-planar normal sections

Abstract

Let M be a smooth m-dimensional submanifold in (m + d)- dimensional Euclidean space Um+d. For x e M and a non-zero vector X in TXM, we define the (d + l)-dimensional affine subspace E(x,X) of Um+d by E(x, X) = x + spw{X, NX(M)}. In a neighbourhood of x, the intersection M DE(x,X) is a regular curve y:(-e , e)—»M. We suppose the parameter f e (-e, e) is a multiple of the arc-length such that y(0) = x and y(0) = X. Each choice of X e T(M) yields a different curve which is called the normal section of M at JC in the direction of X, where X s TX(M) (Section 3). For such a normal section we can write y(t) = x + \(t)X + N(t). (0.1) where N(t) e NX(M) and A(f) e R. The submanifold M is said to have pointwise k-planar normal sections (Pk-PNS) if for each normal section y the first, second and higher order derivatives are linearly dependent as vectors in Mn+d. Submanifolds with pointwise 3-planar normal sections have been studied by S. J. Li in the case when M is isotropic [6] and also in the case when M is spherical [7]. In this paper we consider product submanifolds M = MXX M2 with P3-PNS and we show that this implies strong conditions on Mj and M2