Complex structures on certain differentiable 4-manifolds

Orthogonal complex structures on certain Riemannian 6-manifolds∗

It is shown that the Hermitian-symmetric space CP1 × CP1 × CP1 and the flag manifold F1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of co...

DIFFERENTIABLE STRUCTURES 1. Classification of Manifolds

(1) Classify all spaces ludicrously impossible. (2) Classify manifolds only provably impossible (because any finitely presented group is the fundamental group of a 4-manifold, and finitely presented groups are not able to be classified). (3) Classify manifolds with a given homotopy type (sort of worked out in 1960’s by Wall and Browder if π1 = 0. There is something called the Browder-Novikov-Su...

0 Cones and causal structures on topological and differentiable manifolds

General definitions for causal structures on manifolds of dimension d + 1 > 2 are presented for the topological category and for any differentiable one. Locally, these are given as cone structures via local (pointwise) homeomorphic or diffeomorphic abstraction from the standard null cone variety in IR d+1. Weak (C) and strong (C m) local cone (LC) structures refer to the cone itself or a manifo...

Notes on Differentiable Manifolds

Complex structures on 4-manifolds with symplectic 2-torus actions

We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a 4-manifold with a symplectic 2-torus action admits an invariant complex structure and give an identification of those that do not admit a Kähler structure with Kodaira’s class of complex surfaces which admit a nowhere vanishing holomorphic (2, 0)-form, but are not a torus or a K3 surface.