We study submanifolds by using the notion of circles in Riemannian
geometry. This expository paper consists of two parts. One is to improve Nomizu and Yano’s result ([11]) which characterizes extrinsic spheres in a Riemannian manifold. The other is to characterize totally geodesic immersions and some parallel immersions in terms of Frenet curves of proper order 2 (which are generalizations of circles) on submanifolds.