On the Roter type of Chen ideal submanifolds (Q545503)

The authors consider Chen ideal submanifolds in Euclidean ambient spaces and their curvature properties of pseudo-symmetry type. Chen ideal submanifolds are submanifolds \(M^n\) of \(\mathbb E^{n+m}\) for which Chen's basic inequality for \(\delta(2)\)-curvature is an equality at all points of \(M^n\). The authors describe necessary and sufficient conditions for a Chen ideal submanifold to be pseudo-symmetric. Roter manifolds are Riemannian manifolds, whose Riemann-Christoffel curvature tensor \(R\) can be expressed as a linear combination of Kulkarni-Nomizu products of the metric tensor \(g\) and the Ricci tensor \(S\). Since for real space forms the tensor \(R\) is proportional to the Kulkarni-Nomizu square of their metric tensor \(g\), Roter manifolds can be seen as Riemannian manifolds whose Riemann-Christoffel curvature tensor \(R\) has the simplest expression after that of real space forms. The authors prove that every Chen ideal submanifold \(M^n\) in \(\mathbb E^{n+m}\) is a Roter manifold if and only if it is pseudosymmetric.