This paper is devoted to \({\overset {1} N}\)-symmetric submanifolds of Euclidean spaces, which are characterized as those having covariantly constant first normal map. (The locally symmetric submanifolds in the sense of D. Ferus form a special case). The main result is the characterization of all \({\overset {1} N}\)-symmetric submanifolds of Euclidean spaces which are totally geodesic submanifolds of direct products \(M\times \mathbb{R}^ d\), where \(M\) is a locally symmetric submanifold of some \(\mathbb{R}^ m\).