# What is a real structure on a manifold?

I have been looking at manifolds (twistor spaces) that have a "real structure". I am not quite sure what this means. I've looked on Wikipedia and they have an article that explains real structures on vector spaces and this makes sense. But I'm not sure how you extrapolate to manifolds. Do you need such a map at every tangent space $T_p M$? Or do you just need an antilinear involution $\sigma: M \rightarrow M$ with some non-empty fixed set?

Also, for the vector space case, what you end up with is $V_{\mathbb{R}} \subset V$ such that $V_{\mathbb{R}} \otimes_{\mathbb{R}} \mathbb{C} \cong V$. This is a nice statement and I can see why you'd like this subspace. But do we have an analogous statement for complex manifolds? Are there such things as complex manifolds without real structures?

So in short, what is a real structure on a complex manifold and why is it useful or important? Thanks for the help.

• It's probably referring to an identification of the bundle $\mathbb{C}^n \to TM \to M$ with $\mathbb{R}^{2n} \to TM \to M$ (globally, not just at a particular fiber). – anomaly Apr 30 '15 at 18:19