On normally flat Einstein submanifolds
Let \(M\) be a Riemannian 4-manifold isometrically immersed with flat normal connection in Euclidean 6-space \(E^6\). In terms of the Weingarten operators \(A_{ N_1}\), \(A_{N_2}\), which can be diagonalized simultaneously, as is well-known, a classification is given for the case of Einstein manifolds \(M\). An example is given showing that the scalar curvature of such an immersed \(M\) can be any given real number (if the codimension is 1, it must be positive!). This classification is then specified for the case of Ricci-flat \(M\). It is shown that an \(n\)-dimensional Riemannian manifold \(M\), isometrically immersed with flat normal connection in \(E^{n+N}\), is flat if and only if it is cylindrical, i.e., if there exists an orthonormal frame field with directions \(\xi\) such that every \(A_\xi\) admits a zero eigenvalue with multiplicity of \(n-1\) or \(n\). The relation with Cartan's conjecture (that there exist flat non-cylindrical submanifolds of dimension \(>3\) in Euclidean spaces) was previously discussed. [See also: \textit{Ü. Lumiste}, Sov. Math. 31, 23-35 (1987; Zbl 0616.53006)].
scientific article; zbMATH DE number 1054255
scientific article