Tensor Contraction - Application To Tensor Fields

Application To Tensor Fields

Contraction is often applied to tensor fields over spaces (e.g. Euclidean space, manifolds, or schemes). Since contraction is a purely algebraic operation, it can be applied pointwise to a tensor field, e.g. if T is a (1,1) tensor field on Euclidean space, then in any coordinates, its contraction (a scalar field) U at a point x is given by

Since the role of x is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors.

Over a Riemannian manifold, a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory. For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor.

One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold or the context of sheaves of modules over the structure sheaf; see the discussion at the end of this article.