A metric tensor g is a symmetric, covariant, rank 2, non-degenerate tensor.  The inverse of a metric tensor h is a symmetric, contravariant, rank 2 non-degenerate tensor.

If X is a vector field or rank 1 contravariant tensor, then we can use the metric g to define a rank 1 covariant tensor field alpha = L_g(X) by alpha(Y) = g(X, Y), where Y is any vector field.  In terms of components, if X = X^i partial_{x^i} and g = g_{ij} dx^i dx^j, then alpha = a_i dx^i where a_i = g_{ij} X^j.  The mapping L_g, called the lowering map, is easily extended to higher rank tensors.  For example, if T(alpha, beta, gamma) is a contravariant rank 3 tensor, then a mixed type S of covariant rank 2 and contravariant rank 1 can be defined by lowering, say the 1st and 3rd indices of T, by setting S(X, beta, Y) = T((L_g(X), beta, L_g(Y)).

The lowering map L_g from rank 1 contravariant tensor fields to rank 1 covariant tensor fields is invertible.  The inverse map from rank 1 covariant tensor fields to rank 1 contravariant tensor fields is called the raising map R_h and is defined in terms of the inverse h of the metric g.  If X = R_h(alpha), then in terms of components X^i = h^{ij} a_j, where [h^{ij}] is the inverse matrix for [g_{ij}].  As in the case of the lowering map, the raising map can be extended to higher rank tensors.

With the first calling sequence, RaiseLowerIndices(g, T, Indices) will use the metric g to lower the indices/arguments of T given in the list Indices.  Each of these indices must be a contravariant index.

With the second calling sequence, RaiseLowerIndices(h, T, Indices) will use the inverse metric h to raise the indices/arguments of T given in the list Indices.  Each of these indices must be a covariant index.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RaiseLowerIndices(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-RaiseLowerIndices.