On submanifolds with planar normal sections (Q1059283)

A theorem of \textit{B. Y. Chen} [J. Geom. 20, 122-127 (1983; Zbl 0518.53005)] is generalized to higher dimensions. We prove the following theorems. Theorem B: Let M be an n-dimensional submanifold in \(E^ m\) with planar normal sections. If, locally, M does not lie in an \((n(n+1)/2)\)- dimensional affine subspace of \(E^ m\), then M is an open portion of a Veronese submanifold \(V^ n\) in an \((n+n(n+1)/2)\)-subspace of \(E^ m.\) Theorem C. Let M be a three-dimensional submanifold in \(E^ m\) with planar normal sections. If, locally, M does not lie in a five-space \(E^ 5\) of \(E^ m\), then M is an open portion of a Veronese submanifold \(V^ 3\) in \(E^ 9\) or is the Riemannian product of the real line R with the Veronese surface.