Parallel submanifolds with an intrinsic product structure

Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\dim(M_i)\geq 2$ for $i=1,...,r$ such that $M\cong M_1\times...\times M_r$ . As a starting point, we describe how the intrinsic product structure of $M$ is reflected by a distinguished, fiberwise orthogonal direct sum decomposition of the corresponding first normal bundle. Then we consider the (second) osculating bundle \osc f, which is a $\nabla^N$-parallel vector subbundle of the pullback bundle $f^*TN$, and establish the existence of $r$ distinguished, pairwise commuting, $\nabla^N$-parallel vector bundle involutions on \osc f . Consequently, the "extrinsic holonomy Lie algebra" of \osc f bears naturally the structure of a graded Lie algebra over the Abelian group which is given by the direct sum of $r$ copies of $\Z/2 \Z$ . Our main result is the following: Provided that $N$ is of compact or non-compact type, that $\dim(M_i)\geq 3$ for $i=1,...,r$ and that none of the product slices through one point of $M$ gets mapped into any flat of $N$, we can show that $f(M)$ is a homogeneous submanifold of $N$ .Comment: 25 pages, Appendix A added, a few corrections, new numbering of the theorem

Submanifolds with parallel second fundamental form studied via the Gauß map

For an arbitrary n-dimensional riemannian manifold N and an integer m between 1 and n-1 a covariant derivative on the Graßmann bundle G_m(TN) is introduced which has the property that an m-dimensional submanifold M of N has parallel second fundamental form if and only if its Gauß map (defined on M with values in G_m(TN)) is affine. (For the case that N is the euclidian space this result was already obtained by J.Vilms in 1972.) By means of this relation a generalization of E. Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing an m-dimensional tangent space W and a second fundamental form b at p in N are given; for these data we construct an m-dimensional ``umbrella'' M=M(p,W,b) in N, the rays of which are helical arcs of N; moreover we present tensorial conditions (not involving the covariant derivative on G_m(TN)) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way