Define the square $d$ dimensional Stiefel manifold as $$V_{d} = \{ R \in \mathbb{R}^{d \times d} : R ^\top R = I_d \} .$$ How does one integrate on this manifold over a domain defined as $\{ R \in V_{d} : \text{tr}(R^\top A) > 0 \}$, where $A \in \mathbb{R}^{d \times d}$? My goal is to analytically quantify the ratio of this volume (intuitively it feels like this should be the surface area of a hypersphere) over the entire domain of $R$, to understand how big/small the size of this support is based on $A$.

What are some good resources to look into?