Abstract. We consider a generalization of the idea of an e-tube about a submanifold of IR ~ which includes, on the one hand, submanifolds parallel to the original and, on the other, isoparametric submanifolds about a focal submanifold. We discuss properties that are inherited from the core manifold and the type fibre. The construction is used to show that there are submanifolds with 'many different' parallel submanifolds.
0. Introduction. Let M be a smooth m-dimensional submanifold in (m + d)-dimensional Euclidean space U m+d . For x e M and a non-zero vector X in T X M, we define theIn 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 T X (M) (Section 3).For such a normal section we can write 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 = M X X M 2 with P3-PNS and we show that this implies strong conditions on Mj and M 2 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.