The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. This isomorphism is obtained by setting, for each tangent vector ,

the linear functional on which sends a tangent vector at to . That is, in terms of the pairing between and its dual space ,

for all tangent vectors and . The mapping is a linear transformation from to . It follows from the definition of non-degeneracy that the kernel of is reduced to zero, and so by the rank–nullity theorem, is a linear isomorphism. Furthermore, is a symmetric linear transformation in the sense that

for all tangent vectors and .

Conversely, any linear isomorphism defines a non-degenerate bilinear form on by means of

This bilinear form is symmetric if and only if is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on and symmetric linear isomorphisms of to the dual .

As varies over , defines a section of the bundle of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as : it is continuous, differentiable, smooth, or real-analytic according as . The mapping , which associates to every vector field on a covector field on gives an abstract formulation of "lowering the index" on a vector field. The inverse of is a mapping which, analogously, gives an abstract formulation of "raising the index" on a covector field.

The inverse defines a linear mapping

which is nonsingular and symmetric in the sense that

for all covectors , . Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map

or by the double dual isomorphism to a section of the tensor product

Canonical measure and volume form
ভূমিকা
Tangent–cotangent isomorphism
The energy, variational principles and geodesics