On the Manifold of Almost Complex Structures

Let $(M,g_0)$ be a smooth closed Riemannian manifold of even dimension $2n$ admitting an almost complex structure. It is shown that the space $\mathscr A^+$ of all almost complex structures on $M$ determining the same orientation as the one determined by a fixed almost complex structure $J_0$ is a smooth locally trivial fiber bundle over the space $\mathscr A\mathscr O_{g_0}^+$ of almost complex structures orthogonal with respect to $g_0$ and determining the same orientation as $J_0$.