A Riemannian metric \(g\) provides an isomorphism \(\hat{g}\) between the tangent bundle \(TM\) and the cotangent bundle \(T^*M\). It is defined by
\begin{equation*} \hat{g}(X)(Y)=g(X,Y). \end{equation*}Because of the coordinate representation \(\hat{g}(X)=g_{ij}X^i dx^j\) we say the covector field \(\hat{g}(X)\) is obtained from \(X\) by lowering an index. Therefore \(\hat{g}(X)\) is denoted by \(X^\flat\) and is called \(X\) flat.
Similarly, we say the vector field \(\hat{g}^{-1}(\omega)\) is obtained from \(X\) by raising an index, since the coordinate representation is \(\hat{g}^{-1}(\omega)=g^{ij}\omega_i \frac{\partial}{\partial x^i}\). It is denoted by \(\omega^{\sharp}\) and called \(\omega\) sharp.
Both notations are from musical notation. That is why \(\hat{g}\) and \(\hat{g}^{-1}\) are called musical isomorphisms.
- \(\hat{g}\) is bijective
- Note, we may also define \(\hat{g}(X)\) for a smooth vector field \(X\). Then \(\hat{g}(X)\) is a smooth covector field and \(\hat{g}\) is a map from \(\mathfrak{X}(M)\) to \(\mathfrak{X}^*(M)\).