Isotropic totally real submanifolds
The authors study n-dimensional totally real isotropic submanifolds of a complex manifold. A submanifold of a Riemannian manifold is called isotropic [\textit{B. O'Neill}, Can. J. Math. 17, 907-915 (1965; Zbl 0171.205)] if \(\| h(v,v)\|^ 2=\lambda (p),\) where h denotes the second fundamental form, is independent of the unit tangent vector v at the point p. If \(\lambda\) is also independent of the point p, then the submanifold is called a constant isotropic submanifold. The main results are the following. (i) Let \(M^ n\) (n\(\geq 3)\) be a minimal, totally real submanifold isometrically immersed in a Kaehler manifold \(\tilde M^ n\). If \(M^ n\) is isotropic, then either \(M^ n\) is totally geodesic or \(n=5,8,14\) or 26. (ii) Every n-dimensional, constant isotropic, totally real submanifold of a complex space form, with constant holomorphic sectional curvature c, \(\tilde M^ n(c)\), where \(c\leq 0\), is totally geodesic. (iii) The authors give a complete classification of complete, constant isotropic, totally real submanifolds of \({\mathbb{C}}P^ n(c)\).
scientific article; zbMATH DE number 4109568