## 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.

### MSC:

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

### Keywords:

complex space form; totally real surface; mean curvature