Infinitesimally homogeneous submanifolds of Euclidean spaces
In 1960, I. M. Singer introduced the concept of an infinitesimally homogeneous Riemannian manifold \((M,g)\). This means that the curvature tensor and its covariant derivatives up to certain order \(k(n) + 1\) are ``the same'' for all points of \(M\); here \(k(n)\) is an integer depending only on \(n = \dim M\) (now usually called the Singer's invariant). The main theorem says that every infinitesimally homogeneous manifold is locally homogeneous. The present author gives an extrinsic analogue of the previous theory. Here the second fundamental form of a submanifold \(M \subset \mathbb{R}^N\) and its covariant derivatives play the corresponding role and an extrinsic Singer's invariant'' is defined. One of the main results says that each ``infinitesimally homogeneous submanifold'' \(M\) of \(\mathbb{R}^N\) is an open part of a globally homogeneous submanifold. An analogue is not true in the ``intrinsic'' theory because a locally homogeneous Riemannian manifold may not be locally isometric to a globally homogeneous one. An example is given for which the intrinsic and the extrinsic Singer's invariant are not equal. Another main result says that the second fundamental form and its covariant derivatives up to order \(k + 1\) (where \(k\) is the extrinsic Singer's invariant) at any fixed point determine the immersion uniquely. The main results of this paper generalize well-known results by D. Ferus and H. Reckziegel on submanifolds with parallel second fundamental form. An application to isoparametric submanifolds is also given.
scientific article; zbMATH DE number 762877