$SL(n,\mathbb{C})\rightarrow GL(2n,\mathbb{R})$ reduction of frame bundle

Question

When we study the frame bundle of a n-dimensional Riemannian manifold $M$ we start with a principal $GL(n,\mathbb{R})$ bundle over $M$. There are a series of topological obstructions to reducing the structure. If the bundle is orientable we have a reduction $$GL^{+}(n,\mathbb{R})\rightarrow GL(n,\mathbb{R}) $$

If the manifold admits a metric (always since we're talking Riemannian), then we get:

$$O(n,\mathbb{R})\rightarrow GL(n,\mathbb{R})$$

We call it a reduction to the orthonormal frame bundle. Suppose we look at instead:

$$SL(n,\mathbb{R})\rightarrow GL(n,\mathbb{R})$$

Then we have a volume form on our manifold. I'm familiar with the idea, what I'm wondering about is a reduction of the form:

$$SL(n,\mathbb{C})\rightarrow SL(2n,\mathbb{R})$$

for a manifold of even dimension. What does that correspond to? What are the topological obstructions to it's existence? Or we might consider instead:

$$GL(n,\mathbb{C})\rightarrow GL(2n,\mathbb{R})$$

is this just the existence of a complex structure, or is there more, what kind of frames does this reduction of the frame bundle correspond to?

Answer 1

Let me answer the $GL$ version of your question (from the last paragraph). Suppose that $P\to M$ is a principal $GL(2n, {\mathbb R})$-bundle over a manifold $M$ (this need not be the frame bundle of the tangent bundle). Let $E\to M$ be the associated ${\mathbb R}^{2n}$-bundle.

I am assuming that $GL(n, {\mathbb C})$ is embedded in $GL(2n, {\mathbb R})$ in the standard fashion: $$ A+iB\mapsto A\oplus B $$ (the image is a block-diagonal matrix). Then each reduction of $P$ to a $GL(n, {\mathbb C})$-bundle just amounts to a choice of a complex structure $J$ on the bundle $E\to M$ (and vice-versa). In the case when $E\to M$ is the tangent bundle over $M$, the complex structure $J$ on $E\to M$ is called an almost complex structure on $M$. (To be a genuine complex structure, it has to satisfy an integrability condition.) In terms of frames, this can be described as follows: Suppose that $J$ is a complex structure on the bundle $E\to M$ and for $x\in M$, $(v_1,...,v_n)$ is a basis in the complex vector space $(E_x,J)$ (the fiber over $x$). Then the corresponding real frame (a real basis in $E_x$) will be $$ (v_1,...,v_n, Jv_1,..., Jv_n). $$ Conversely, a real frame $$ (v_1,...,v_n, w_1,...,w_n) $$ comes from a complex frame if and only if $w_k=Jv_k$, $k=1,...,n$.

If you really want to understand the $SL$-case, you will have to deal with a complex volume form $\omega\in \Omega^{n,0}(M)$ (nondegenerate at every point $x\in X$). Then complex frames $(v_1,...,v_n)$ will be required to satisfy $$ \omega(v_1,...,v_n)=1. $$ Since you did not explain a motivation for this, I will stop here.