My problem at hand is the construction of a Brownian motion on a constraint matrix manifold$^\color{magenta}\star$ denoted by $\mathcal{M}$, which is represented as follows:
$$ \mathcal{M} = \left\{ P\in \mathbb{R}^{n\times d} \mid \operatorname{Tr} \left( P^T A_k P \right) = b_k, k = 1, 2, \dots, m \right\} $$
where $A_k \in \mathbb{R}^{n\times n}$ are given symmetric matrices and $b_k \in \mathbb{R}$ are given scalars. My initial thought is to construct this Brownian motion based on the tangent space of the manifold. The tangent space at a point $P$ on the manifold, denoted by $T_{P}\mathcal{M}$, is defined as follows:
$$ T_{P}\mathcal{M} = \left\{ Z \in \mathbb{R}^{n\times d} \mid \operatorname{Tr} \left( P^T A_k Z \right) = 0, k = 1, 2, \dots, m \right\} $$
However, obtaining the basis of this tangent space poses a challenge for me as I didn't find explicit forms of those bases. Does anyone have an idea about constructing a Brownian motion for this particular manifold?
Another thought is to project the Brownian motion from space of all matrices $\mathbb{R}^{n\times d}$ to this manifold. As the normal space w.r.t. Frobenius inner product is
$$ N_{P}\mathcal{M} = \left\{\sum_{k=1}^m \alpha_k A_k P, \alpha_k \in \mathbb{R}\right\}, $$
the orthogonal projection to $T_P \mathcal{M}$ is $\pi_P(X)= X - \sum_{k=1}^m \alpha_k A_k P$ and these $\alpha_k$ can be solved by equations $\operatorname{Tr} \left( P^T A_k X \right) = 0, k = 1, 2, \dots, m$. However, I cannot connect the relationship between the projected Brownian motion and the Brownian motion on this manifold as they are not the same in general.
$\color{magenta}\star$ Michel Journée, Francis Bach, Pierre-Antoine Absil, Rodolphe Sepulchre, Low-rank optimization on the cone of positive semidefinite matrices, SIAM Journal on Optimization, Volume 20, Issue 5, 2010.