Totally umbilic isometric immersions and curves of order two (Q870652)

This paper studies curves of variable geodesic curvatures on submanifolds. A condition is given that these submanifolds are totally umbilic in terms of extrinsic shapes of such curves. It is also shown that under this condition their non-totally geodesic parts have a parallel normalized mean curvature vector and that they are essentially conformally equivalent to extrinsic spheres when they are embeddings. This gives an extension of Nomizu-Yano's result on a characterization of extrinsic spheres.