On submanifolds of Hopf manifolds (Q1105847)

Let \(H^ m\) be diffeomorphic to \(S^{2m-1}\times S^ 1\) with the complex structure given by Hopf, called a Hopf manifold [see \textit{S. Kobayashi} and \textit{K. Nomizu}, Foundations of differential geometry, Vol. II (1969; Zbl 0175.485), p. 137]. A submanifold M of \(H^ m\) is said to have semi-flat normal connection if the normal curvature tensor \(R^{\perp}\) satisfies: \(R^{\perp}(X,Y)\xi =\rho \circ g(X, \tan JY) nor J\xi\) for some real-valued smooth function \(\rho\) on M any tangent vector fields X, Y on M and any normal section \(\xi\), where tan Z and nor Z denote the orthogonal projection of Z on the tangent and on the normal bundle respectively, and J denotes the complex structure of \(H^ m.\) It is said to be a quasi-Einstein submanifold if the Ricci tensor of M satisfies: \(Ric=ag+bw\otimes w\) for some real-valued smooth functions a, b on M, where w denotes the 1-form naturally induced on M by the Lee form of \(H^ m\). In this article it is proved that if M is an invariant submanifold of \(H^ m\) with semi-flat normal connection then it is a totally umbilical quasi-Einstein submanifold with flat normal connection unless it a complex hypersurface. It is also proved that the only totally-umbilical invariant submanifolds of zero scalar curvature of \(H^ m\) are the totally-geodesic flat surfaces.