# Integrating over some domain of the Stiefel manifold to analyze its support

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?

• your $V_d$ is the orthogonal group. If $R \in V_d$ then also $-R\in V_d$ and exactly one of them satisfies the inequality, so the ratio is 50%. – user35593 Apr 1 '19 at 3:39
• except of course if $trace(R^TA)=0$ but this set has measure except if $A=0$. – user35593 Apr 1 '19 at 6:54
• On orthogonal groups, compact lie groups and their homogeneous spaces one typically uses the bi-invariant measure for integration. Your decomposition using the trace is basically the decomposition of $O_{d-1}$ into $SO_{d-1}$ and its complement, if I understand your conventions. – Ryan Budney Apr 1 '19 at 15:25