An inequality for totally real surfaces in complex space forms. (English) Zbl 1080.53052

Let \(M\) be a totally real surface in a complex space form \(\widetilde M(4c)\) of arbitrary codimension, where \(c\) is the holomorphic sectional curvature. If \(H\) is the mean curvature vector of \(M\), \(K\) is the Gauss curvature, and \(K^E\) the elliptic curvature of \(M\), then the inequality \(\| H\| ^2\geq K-K^E-c\) is proved.
Using the notion of ellipse of curvature, a characterization of the equality in this inequality is also obtained. An example of a Lagrangian surface of the space \(\mathbb C^2\) satisfying the equality is given.


53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds