Existence and uniqueness theorem for slant immersions and its applications (Q678724)

A slant isometric immersion \(f:M\to\widetilde{M}\) is an isometric immersion from a Riemannian manifold \(M\) into an almost Hermitian manifold \(\widetilde{M}\) with constant Wirtinger angle \(\theta\), i.e., for every \(p\in M\) and every unit vector \(v\in T_pM\) the angle between \(Jf_*v\) and \(f_*T_pM\) is \(\theta\). The authors prove an existence and uniqueness theorem for slant isometric immersions from an \(n\)-dimensional Riemannian manifold \(M\) into a complex space form \(\widetilde{M}\) of complex dimension \(n\) and some constant holomorphic curvature. This result is along the lines of the analogous theorem on isometric immersions into real space forms. In this article the assumptions (related to the curvature equations of Gauss, Codazzi and Ricci) are expressed by entities on \(M\) which reflect the complex structure \(J\) of the ambient space. For \(n=2\) special existence results are derived, for instance on slant surfaces with constant mean curvature.