On the Roter Type of Chen Ideal Submanifolds

Abstract

Chen ideal submanifolds Mnin Euclidean ambient spaces En+m (of arbitrary dimensions n ≥ 2 and codimensions m ≥ 1) at each of their points do realise an optimal equality between their squared mean curvature, which is their main extrinsic scalar valued curvature invariant, and their δ-(= δ(2)-) curvature of Chen, which is one of their main intrinsic scalar valued curvature invariants. From a geometric point of view, the pseudo-symmetric Riemannian manifolds can be seen as the most natural symmetric spaces after the real space forms, i. e. the spaces of constant Riemannian sectional curvature. From an algebraic point of view, the Roter manifolds can be seen as the Riemannian manifolds whose Riemann-Christoffel curvature tensor R has the most simple expression after the real space forms, the latter ones being characterisable as the Riemannian spaces (Mn, g) for which the (0, 4) tensor R is proportional to the Nomizu-Kulkarni square of their (0, 2) metric tensor g. In the present article, for the class of the Chen ideal submanifolds Mn of Euclidean spaces En+m, we study the relationship between these geometric and algebraic generalisations of the real space forms. © 2011 Springer Basel AG.