On Chen immersions into Lorentzian space forms with nonflat normal space (Q2714301)

Let \(f:M^m_p\to\tilde M^{m+n}_q\) be a smooth immersion. Let \(\xi\) be a nonzero and nonnull vector in the normal bundle and choose an orthonormal local reference \(\{\xi_1,\xi_2,\dots,\xi_n\}\) in \(T^\perp M\) such that \(\xi_1=\xi/\|\xi\|\). The allied vector field \({\mathcal A}(\xi)\) of \(\xi\) is defined by \({\mathcal A}(\xi)=\sum_{i=2}^n\epsilon_i\text{trace}(A_\xi A_{\xi_i})\xi_i\), where \(\epsilon_i=\pm 1\). In particular, \({\mathcal A}(H)\) is called the allied mean curvature vector of \(M\) in \(\tilde M\). The immersion \(f\) is said to be an \({\mathcal A}\)-immersion or a Chen-immersion if \(H=0\) or \(\langle H,H\rangle\neq 0\) and \({\mathcal A}(H)=0\). In this paper the author takes a totally geodesic isometric immersion from an \(m\)-dimensional connected Riemannian manifold into an \((m+2)\)-dimensional Lorentzian space form and constructs examples of Chen immersions with nonflat normal bundle.