# Parallel Submanifolds of Space Forms

• Masaru Takeuchi
Chapter
Part of the Progress in Mathematics book series (PM, volume 14)

## 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 IRm or a symmetric R-space M ∈ ℝm, which is minimally imbedded in a hypersphere of IRm (cf. Takeuchi-Kobayashi [12]), is a parallel submanifold of IRm. Ferus ([3], [4]) showed that essentially these submanifolds exhaust all parallel sub-manifolds of ℝm in the following sense: A complete full parallel submanifold of the Euclidean space IRm = Mm (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 Mi ⊂ ℝ is an irreducible symmetric R-space.

## Keywords

Vector Field Riemannian Manifold Fundamental Form Space Form Isometric Immersion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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