Isoparametric families of submanifolds (Q5918514)

This article is a translation into English of the article [Bol. Soc. Bras. Mat. 13, 35--48 (1982)] originally published in Portuguese. This paper is a study of isoparametric families of submanifolds of arbitrary codimension in a Riemannian manifold. Among the main results, it is shown that the mean curvature vector belonging to a submanifold in an isoparametric family has constant length. Furthermore, if the orthogonal distribution to the isoparametric family is integrable, then the integral manifolds of the distribution are totally geodesic and the mean curvature vector belonging to a submanifold in the isoparametric family is parallel. Moreover, when the ambient Riemannian manifold has constant curvature, the submanifolds of the isoparametric family have constant principle curvatures which satisfy a formula that generalizes Cartan's formula for the principle curvatures in the codimension 1 case [\textit{ Cartan}, Ann. Mat. Pura Appl. (4) 17, 177--191 (1938; Zbl 0020.06505)].