What is a real structure on a manifold?

Question

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.

Answer 1

The definition I know can be phrased for any complex vector bundle, and it is a real structure on each fibre, varying smoothly along the manifold.

For a complex manifold, you can of course just take the tangent bundle. But if you have e.g. a real spin manifold, you can adjoin some complex representation of the spin group and get a complex spinor bundle. Real structures on these bundles are quite important objects e.g. in the study of noncommutative geometry: Connes proved that if you start with a Riemannian manifold, take the Dirac operator, the *-algebra of functions, the spinor bundle and its real structure (and lots of technical details), you can recover the Riemannian manifold from it.

If you have a real structure on a complex vector bundle in the sense explained above, you have a real structure on the (vector) space of sections of the vector bundle, in the way you already know.