Submanifolds with geodesic normal sections
Let m be a submanifold of a Euclidean m-space \(E^ m\). For each point \(p\in M\) and a unit tangent vector t at p, the normal space of M at p and t determine an affine subspace E(p,t) which intersects M along a curve in a neighborhood of p, called the normal section at p in the direction t. Since M is Riemannian with respect to the induced metric, there is another canonical curve on M, the geodesic, through p with t as its tangent at p. In this paper, the authors ask and study the following geometric question: ''which are the submanifolds of \(E^ m\) whose normal sections are geodesics?'' The authors give many examples of submanifolds of \(E^ m\) with geodesic normal sections which include all ''standard immersions'' of strongly harmonic spaces. They prove that every submanifold with geodesic normal sections is in fact constant isotropic. Furthermore, two characterization theorems together with several other fundamental results for such submanifolds are obtained. Moreover, they also obtain some classification theorems for surfaces with geodesic normal sections. As a consequence, a new characterization of the Veronese surface in terms of geodesic normal sections is derived. [Remark. Submanifolds with ''simple'' geodesics are also studied very recently by \textit{D. Ferus} and \textit{S. Schirrmacher} [Math. Ann. 260, 57-62 (1982; Zbl 0474.53004)] and \textit{K. Sakamoto\&(Math. Ann. 261, 63-80 (1982; Zbl 0487.53054).] }
scientific article; zbMATH DE number 3851964