Center of mass and Kähler structures

Abstract

There is a sequence of positive numbers ${\delta }_{2n}$, such that for any connected $2n$-dimensional Riemannian manifold $M$, there are two mutually exclusive possibilities: $1)$ There is a complex structure on $M$ making it into a Kähler manifold, or $2)$ For any almost complex structure $J$ compatible with the metric, at every point $p\in M$, there is a smooth loop $\gamma$ at $p$ such that $dist(J_p,ho{l}_{\gamma }^{-1}{J}_{p}ho{l}_{\gamma })>{\delta }_{2n}$.

4 pages