stylistic:style_transfer
Raising and lowering indices

In a basis of vector fields f = (X,..., X), any smooth tangent vector field X can be written in the form

for some uniquely determined smooth functions v,..., v. Upon changing the basis f by a nonsingular matrix A, the coefficients v change in such a way that equation (7) remains true. That is,

Consequently, v = A v. In other words, the components of a vector transform contravariantly (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix A. The contravariance of the components of v is notationally designated by placing the indices of v in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields f = (X,..., X) define the dual basis to be the linear functionals (θ,..., θ) such that

That is, θ (X) = δ, the Kronecker delta. Let

Under a change of basis f ↦ f A for a nonsingular matrix A, θ transforms via

Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ

where a denotes the row vector... a ]. The components a transform when the basis f is replaced by f A in such a way that equation (8) continues to hold. That is,

whence, because θ = A θ, it follows that a = a A. That is, the components a transform covariantly (by the matrix A rather than its inverse). The covariance of the components of a is notationally designated by placing the indices of a in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding X fixed, the function

of tangent vector Y defines a linear functional on the tangent space at p. This operation takes a vector X at a point p and produces a covector g (X, −). In a basis of vector fields f, if a vector field X has components v, then the components of the covector field g (X, −) in the dual basis are given by the entries of the row vector

Under a change of basis f ↦ f A, the right-hand side of this equation transforms via

so that a = a A: a transforms covariantly. The operation of associating to the (contravariant) components of a vector field v = v... v ] the (covariant) components of the covector field a = a … a ], where

is called lowering the index.

To raise the index, one applies the same construction but with the inverse metric instead of the metric. If a = a... a ] are the components of a covector in the dual basis θ, then the column vector

has components which transform contravariantly:

Consequently, the quantity X = f v does not depend on the choice of basis f in an essential way, and thus defines a vector field on M. The operation (9) associating to the (covariant) components of a covector a the (contravariant) components of a vector v given is called raising the index. In components, (9) is