I've been studying Riemannian geometry from John M. Lee's book on the same. My aim is to understand the geometry of Hermitian positive semidefinite matrices when viewed as a Riemannian manifold.
In Lee's book (and Wikipedia), the Riemannian metric (tensor) $g$ is defined as a smooth 2-covariant tensor field on a manifold $M$. Equivalently, for every point $p \in M$, it smoothly assigns an 2-covariant tensor (which can be interpreted as an inner product) of the form
$$ g(p) \equiv g_p : T_pM \times T_pM \to \mathbb{R}. $$
Due to the inner product nature, one can also denote it as
$$ g_p(X,Y) \equiv \langle X, Y \rangle_p, $$ for $X, Y \in T_pM$.
This is a definition of the Riemannian tensor I'm comfortable with.
However, in the book 'Positive Definite Matrices' by Rajendra Bhatia, he defines the metric via a differential. Here the manifold is the set of $n \times n$ Positive definite matrices $ \mathbb P_n$, and for a point $A \in \mathbb P_n$, the tangent space is $T_A \mathbb P_n = {A} \times \mathbb H_n \cong \mathbb H_n$, where $\mathbb H_n$ is the space of $n \times n$ Hermitian matrices.
The author states that the inner-product on $ \mathbb H_n$ (defined as $\langle A, B \rangle := \operatorname{Tr}[A^*B]$) leads to a Riemannian metric on the manifold $ \mathbb P_n$. He then says that 'at a point $A$ this metric is given by the differential'
$$ ds = || A ^{-1/2} dA A ^{-1/2} ||_2 = (\operatorname{Tr}[(A ^{-1} d A)^2])^{1/2}, $$
where the $|| \cdot ||$ denotes the Hilbert-Schmidt norm defined as $||A||_2 = \sqrt{\langle A,A \rangle }$.
What is the equivalence between Bhatia's definition and Lee's definition of the metric tensor? What is the right way of interpreting differential here ($ds$ and $dA$)? The interpretation of 'differential' I'm comfortable with is the way John M. Lee defines it, where the differential of a smooth function is a particular kind of covector field.
PS. Am I missing any keyword which would help with looking up for references? I think this has to do induced metric, but I'm not sure.
${\rm d}s$ is a standard notation for the line element ${\rm d}s = (g_{ij}\,{\rm d}x^i\,{\rm d}x^j)^{1/2}$, while $A$ is also used to denote a "global coordinate" on $\mathbb{P}_n$. It is $g_A(B,B)^{1/2} = \|A^{-1/2}BA^{-1/2}\|_2 = \big({\rm Tr}[(A^{-1}B)^2]\big)^{1/2}$ whenever $A\in \mathbb{P}_n$ and $B\in T_A\mathbb{P}_n$. Then polarize this formula to find $g_A(B,C)$ for $B,C\in T_A\mathbb{P}_n$.