10

Let $M$ be an $2n$-dimensional manifold. Let $\mathcal{F}_{\mathrm{GL}(2n, \mathbb{R})}$ be the frame bundle over $M$. Consider the subgroup $\mathrm{GL}(n, \mathbb{C})\subset\mathrm{GL}(2n, \mathbb{R})$. What I'm trying to prove is:

If $M$ has an almost complex structure $J:TM\rightarrow TM$ then there is a reduction of the structure group $\mathrm{GL}(2n, \mathbb{R})$ of $\mathcal{F}_{\mathrm{GL}(2n, \mathbb{R})}$ to $\mathrm{GL}(n, \mathbb{C})$.

The reciprocal is also true, and I was able to prove it. But I'm stuck on this direction. Does anyone have a suggestion?

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Leonardo Schultz
  • 675

1 Answers1

4

Let us interpret the points in the frame bundle over $x\in M$ as linear isomorphisms from $u:\mathbb R^{2n}\to T_xM$. Fixing an identification of $\mathbb R^{2n}$ with $\mathbb C^n$, you can view each $u$ as a real linear isomorphism $\mathbb C^{n}\to T_xM$. Now over $x\in M$For a point $x\in M$, consider the subset of those linear isomorphisms for which $u(iz)=J_x(u(z))$ for all $z\in\mathbb C^n$. It is clear that for such an isomorphism $u$ and $A\in GL(2n,\mathbb R)$ (which here has to be viewed as the group of real linear isomorphisms from $\mathbb C^n$ to itself, $u\circ A$ lies in the subset if and only if $A$ lies in the subgroup $GL(n,\mathbb C)$. Taking a local smooth section $\sigma$ of the frame bundle you can construct a section having values in the suspace in each fiber via $\tilde\sigma(x)(z)=\tfrac12(\sigma(x)(z)-J_x(\sigma(x)(iz))$. Thus we have defined a reduction of structure group.

Andreas Cap
  • 22,380