Parallel Submanifolds of Space Forms

Abstract

A submanifold M of a Riemannian manifold M is said to be parallel if the second fundamental form of $\overline M$ is parallel. For example, an affine subspace M of IR^m or a symmetric R-space M ∈ ℝ^m, which is minimally imbedded in a hypersphere of IR^m (cf. Takeuchi-Kobayashi [12]), is a parallel submanifold of IR^m. Ferus ([3], [4]) showed that essentially these submanifolds exhaust all parallel sub-manifolds of ℝ^m in the following sense: A complete full parallel submanifold of the Euclidean space IR^m = M^m (0) is congruent to

$$M = I{R^{{m_0}}} \times {M_l} \times ...{M_s} \subset I{R^{{m_0}}} \oplus I{R^{{m_1}}} \oplus ... \oplus I{R^{{m_s}}} = I{R^m},\;m = {m_0} + \sum {{m_i}} ,s \ge 0,\;or\;to$$

((a))

so ⩾ 0, or to

$$M = {M_l} \times ... \times {M_s} \subset I{R^{{m_l}}} \oplus ... \oplus I{R^{{m_s}}} = I{R^m},m = \sum {{m_i},s \ge 1,}$$

((b))

where each M_i ⊂ ℝ is an irreducible symmetric R-space.