A symmetric positive definite (not semi-definite) smooth covariant 2-tensor field \(g\) on a manifold \(M\) is called Riemannian metric, i.e. for \(X,Y\in \mathfrak{X}(M)\) the function \(g(X,Y)\) is smooth.
If \(p\in M\) and \(v,w\in T_pM\) we usually write
\begin{equation*} \langle v, w\rangle_g:=g_p(v,w). \end{equation*}