An immersion of an \(n\)-dimensional real space form into an \(n\)-dimensional complex space form (Q1085815)

Let \(M\) be an \(n\)-dimensional submanifold of an \(m\)-dimensional Riemannian manifold \(N\) and \(H\) be the mean curvature vector of \(M\). Let \(\zeta_ x\) be \(m-n\) mutually orthogonal unit normal vector fields of \(M\) such that \(H=| H|_{\zeta_ 1}\). The normal vector \(a(H)\) defined by \[ a(H)=(| H| /n)\sum^{m-n}_{x=2}Tr(A_ 1 A_ x)\zeta_ x \] is called the allied mean curvature vector of \(M\) in \(N\). Chen defined this vector field and called \(M\) an \({\mathcal A}\)-submanifold of \(N\) if \(a(H)=0\) [\textit{B. Y. Chen}, Geometry of submanifolds. Pure and Applied Mathematics. 22. New York: Marcel Dekker (1973; Zbl 0262.53036), Chapter 6]. Many authors studied \({\mathcal A}\)-submanifolds and subsequently they were called Chen submanifolds. The present author studies an \(n\)-dimensional totally real Chen submanifold with constant sectional curvature \(c\) isometrically immersed in an \(n\)-dimensional complex space form \(\tilde M(4\tilde c)\) with \(\tilde c>c\). (M is totally real in \(\tilde M,\) by definition, if the tangent space of \(M\) is mapped into the normal space by the complex structure \(J\).) The author proves a theorem which completely determines such Chen submanifolds. She uses a suitable orthonormal frame on \(M\) and derives the Gauss equations and Codazzi equations. Then the construction of an immersion is shown. This paper gives an example of a local immersion of \(H^ n(-1)\) into an \(n\)-dimensional complex Euclidean space \(\mathbb C^ n\) as a totally real submanifold.