Totally Umbilic Isometric Immersions and Curves of Order Two

In this paper we study submanifolds by use of extrinsic shapes of some curves having points of proper order 2, and give a condition that they are totally umbilic. This gives an extension of Nomizu-Yano’s result in 7] on a characterization of extrinsic spheres. The second author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 17540072), Ministry of Education, Science, Sports, Culture and Technology.