Let \(M\) be an \(n\)-dimensional submanifold of a Euclidean \(m\)-space \(E^ m\). For a unit tangent vector t at a point \(p\) in \(M\), the vector \(t\) and the normal space of \(M\) at \(p\) determine an \((m-n+1)\)-dimensional vector space \(E(p,t)\) in \(E^ m\). The intersection of \(M\) and \(E(p,t)\) gives rise to a curve \(\sigma(s)\) in a neighborhood of \(p\), called the normal section at \(p\) in the direction \(t\). \(A\) submanifold \(M\) is said to have pointwise planar normal sections if each normal section \(\sigma\) at \(p\) satisfies \(\sigma'\wedge \sigma'' \wedge \sigma'''=0\) at \(p\) for each \(p\) in \(M\). The authors classify isotropic submanifolds in \(E^ m\) with pointwise planar normal sections and show that such submanifolds are either open portions of linear subspaces or open portions of a compact rank one symmetric space imbedded in \(E^ m\) by its first standard imbedding.