Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions

http://dx.doi.org/10.4153/CJM-2009-034-4
Canad. J. Math. 61(2009), 641-655

  Published:2009-06-01
 Printed: Jun 2009

For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form $\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we show that if the mean curvature vector of $M^n$ is parallel and the sectional curvature $K$ of $M^n$ satisfies some inequality, then the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is parallel and our manifold $M^n$ is a space form.