Let (M, g) be a Riemannian manifold. Suppose {∂_{i}} is a local frame for the tangent bundle TM with dual coframe {dx^{i}}. Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as g = g_{ij} dx^{i} ⊗ dx^{ j} (where we employ the Einstein summation convention). Given a vector field X = X^{ i}∂_{i} we define its flat by

This is referred to as "lowering an index". Using the traditional diamond bracket notation for inner product defined by g, we obtain the somewhat more transparent relation

for all vectors X and Y.

Alternatively, given a covector field ω = ω_{i}dx^{i} we define its sharp by

where g^{ij} are the elements of the inverse matrix to g_{ij}. Taking the sharp of a covector field is referred to as "raising an index". In inner product notation, this reads

for ω an arbitrary covector and Y an arbitrary vector.

Through this construction we have two inverse isomorphisms

These are isomorphisms of vector bundles and hence we have, for each p in M, inverse vector space isomorphisms between T_{p}M and T ∗ pM.

The musical isomorphisms may also be extended to the bundles

It must be stated which index is to be raised or lowered. For instance, consider the (2, 0) tensor field X = X_{ij} dx^{i} ⊗ dx^{ j}. Raising the second index, we get the (1, 1) tensor field

Trace of a tensor through a metric[edit]

Given a (2, 0) tensor field X = X_{ij} dx^{i} ⊗ dx^{ j} we define the trace of X through the metric g by

Observe that the definition of trace is independent of the choice of index we raise since the metric tensor is symmetric.