High Quality Content by WIKIPEDIA articles! In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or Minkowski metric), one can raise or lower indices: change a type (k, l) tensor to a (k + 1, l – 1) tensor (raise index) or to a (k – 1, l + 1) tensor (lower index). Where the notation (k, l) has been used to denote a rank k + l with k upper indices and l lower indices.One does this by multiplying by the covariant or contravariant metric tensor and then contracting (simply summing over the repeated index j in the example below).

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