I'm still learning Riemannian geometry, so please correct any mistakes.
I am interested in the Bures-Wasserstein manifold of centered Gaussians of dimension $d$. In this case, the manifold $\mathcal{M}$ is the set of $d \times d$ positive definite matrices.
The Riemannian metric on this manifold is defined as $g_P(U,V) = \text{Tr}[L_P(U) P L_P(V)]$, for tangent vectors $U, V \in T_P \mathcal{M}$ where $L_P(U)$ is the solution to the equation $$ L_P(U) P + P L_P(U) = U $$ The exponential map of this manifold is defined as $$ \text{Exp}^{\mathcal M}_P(V) = (\mathbb I + L_P(V)) P (\mathbb I + L_P(V)) $$
Let $\mathcal{D} = \{ P \in \mathcal M : \text{Tr}[P] = 1 \} $ be the set of density matrices.
Does the set of density matrices form a Riemannian submanifold of $\mathcal M$? If so, is the exponential map on this manifold at a point $P \in \mathcal D$ defined as
$$ \text{Exp}^{\mathcal D}_P = \Pi \circ \text{Exp}^{\mathcal M}_P, $$ where $\Pi(X) = X/\text{Tr}[X]$ is the projection onto the set $\mathcal D$ with respect to the Bures distance.
More generally, what is the way of determining if a subset of a manifold forms a submanifold, and is there a way to inherit the exponential map from the parent manifold? Answers specific to the Bures-Wasserstein manifold would also be sufficient.
