In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle ${\displaystyle \mathrm {T} M}$ and the cotangent bundle ${\displaystyle \mathrm {T} ^{*}M}$ of a Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. These isomorphisms are global versions of the canonical isomorphism between an inner product space and its dual. The term musical refers to the use of the musical notation symbols ${\displaystyle \flat }$ (flat) and ${\displaystyle \sharp }$ (sharp).

In the notation of Ricci calculus and mathematical physics, the idea is expressed as the raising and lowering of indices. Raising and lowering indices are a form of index manipulation in tensor expressions.

In certain specialized applications, such as on Poisson manifolds, the relationship may fail to be an isomorphism at singular points, and so, for these cases, is technically only a homomorphism.

In linear algebra, a finite-dimensional vector space is isomorphic to its dual space (the space of linear functionals mapping the vector space to its base field), but not canonically isomorphic to it. This is to say that given a fixed basis for the vector space, there is a natural way to go back and forth between vectors and linear functionals: vectors are represented in the basis by column vectors, and linear functionals are represented in the basis by row vectors, and one can go back and forth by transposing. However, without a fixed basis, there is no way to go back and forth between vectors and linear functionals. This is what is meant by that there is no canonical isomorphism.

On the other hand, a finite-dimensional vector space ${\displaystyle V}$ endowed with a non-degenerate bilinear form ${\displaystyle \langle \cdot ,\cdot \rangle }$ is canonically isomorphic to its dual. The canonical isomorphism ${\displaystyle V\to V^{*}}$ is given by

The non-degeneracy of ${\displaystyle \langle \cdot ,\cdot \rangle }$ means exactly that the above map is an isomorphism. An example is where ${\displaystyle V=\mathbb {R} ^{n}}$ and ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the dot product.

In a basis ${\displaystyle e_{i}}$, the canonical isomorphism above is represented as follows. Let ${\displaystyle g_{ij}=\langle e_{i},e_{j}\rangle }$ be the components of the non-degenerate bilinear form and let ${\displaystyle g^{ij}}$ be the components of the inverse matrix to ${\displaystyle g_{ij}}$. Let ${\displaystyle e^{i}}$ be the dual basis of ${\displaystyle e_{i}}$. A vector ${\displaystyle v}$ is written in the basis as ${\displaystyle v=v^{i}e_{i}}$ using Einstein summation notation, i.e., ${\displaystyle v}$ has components ${\displaystyle v^{i}}$ in the basis. The canonical isomorphism applied to ${\displaystyle v}$ gives an element of the dual, which is called a covector. The covector has components ${\displaystyle v_{i}}$ in the dual basis given by contracting with ${\displaystyle g}$:

This is what is meant by lowering the index. Conversely, contracting a covector ${\displaystyle \alpha =\alpha _{i}e^{i}}$ with the inverse of ${\displaystyle g}$ gives a vector with components