A Riemannian metric $g$ on $M$, n dimensional manifold, is a smooth family of inner products on the tangent spaces of $M$. Namely, $g$ associates to each $p\in M$ a positive definite symmetric bilinear form on $T_p M$, $$g_p:T_p M\times T_p M\to\mathbb R$$ and the smoothness condition on g refers to the fact that the function $$p\in M \to g_p(X_p,Y_p)\in\mathbb R$$ must be smooth for every locally defined smooth vector fields $X, Y$ in $M$.
when I was studying the subject, I wondered what happens if my metric, is of the form $g_p(\xi,\eta)=\xi \cdot A\cdot \eta$, where $A$ is a positive symmetric matrix. How can find the geodesic from this metric? And what happens with a matrix $A=A(x)$?