The measure of holomorphicness of a real submanifold of an almost Hermitian manifold

Abstract:

In this note we define the measure of holomorphicness $\mu (M)$ of a compact real submanifold $M$ of an almost Hermitian manifold $(\overline {M},\overline {J},\overline {g})$. The number $\mu (M)\in [0,1]$ verifies the following properties: $M$ is a complex submanifold iff $\mu (M)=1$; if $\dim M$ is odd, then $\mu (M)=0$. Explicit examples of surfaces in ${\mathbb C}^{2}$ are obtained, showing that $\mu (S^{2})=\frac {1}{5}$ and that $0\leq \mu (T)\leq \frac {3}{8}$, $T$ being the Clifford torus.