The Riemannian metric is usually introduced as a "type" of inner product on the tangent space of a manifold $M$. Furthermore, if $M$ has dimension $n$ it is also known that its tangent space at a point $p \in M$ is a vector space $T_pM$ of dimension $n$. So we have an isomorphism $T_pM \cong \mathbb{R}^n$ and we have the usual inner product on $\mathbb{R}^n$, say $(\cdot,\cdot)$, making it an unitary space.
Let the Riemannian metric be given by $$\langle \cdot,\cdot\rangle_p : T_pM \times T_pM \rightarrow \mathbb{R},$$which is symmetric and positive definite. Now, this is nothing more than a symmetric form and we can obviously associate an operator via the Riesz-Representation theorem, $A : T_pM \rightarrow T_p^*M$ where fixing $v \in T_pM$ we have $Av = f$ and $$T_p^*M \ni Av =: \langle v ,\cdot \rangle_p : T_pM \rightarrow \mathbb{R},$$where the action is given by $$Av(u) = \langle v,u \rangle_p = (f,u), \ \ \forall u \in T_pM.$$
Now, in this way we could define as many metrics as we wanted for each element of $\mathbb{R}^n$. Ok, now the question:
What does the eigenvalues of the metric tell us? And the trace? Is there some geometric information to extract from it? For instance in the Euclidean case, the metric is the Identity matrix, so all the eigenvalues are $1$ and its trace is $n$. In polar coordinates we have the eigenvalues $1$ and $r^2$ and the trace is $1 +r^2$.
Thank you!