# Tensor field

A tensor field (also imprecise tensor) is examined in the mathematical sub-area of differential geometry, in particular in tensor analysis. It is a function that assigns a tensor to each point of an underlying space in a special way .

## definition

Let be a smooth manifold and an (r, s) - tensor bundle . An (r, s) -tensor field is a smooth cut in the tensor bundle . The number of tensor fields is denoted by. This set is a module on the algebra of smooth functions . ${\ displaystyle M}$ ${\ displaystyle T_ {s} ^ {r} (M)}$ ${\ displaystyle T_ {s} ^ {r} (M)}$ ${\ displaystyle \ Gamma ^ {\ infty} (T_ {s} ^ {r} (M))}$ ${\ displaystyle C ^ {\ infty} (M) = \ Gamma ^ {\ infty} (T_ {0} ^ {0} (M))}$ ## Examples

Let M be a differentiable manifold, then a tensor field on M is a mapping that assigns a tensor to every point.

• Riemannian metrics are (0.2) tensor fields.
• The Riemann curvature tensor is a (1,3) -tensor field, which can be interpreted as a (0,4) -tensor field with the help of the Riemannian metric.
• Differential forms of degree k, in particular the total differential of a function in the case k = 1, are sections of where the cotangential bundle is called . For more information, see also Outer Algebra .${\ displaystyle \ textstyle \ bigwedge ^ {k} \ mathrm {T} ^ {*} M \ subseteq (\ mathrm {T} ^ {*} M) ^ {\ otimes k}.}$ ${\ displaystyle T ^ {*} M}$ • The energy-momentum tensor and the electromagnetic field strength tensor (as an example of a field strength tensor) in the theory of relativity are tensor fields of the second order on the four-dimensional basis of Minkowski space .${\ displaystyle T ^ {\ alpha \ beta}}$ ${\ displaystyle F ^ {\ alpha \ beta}}$ • the spin group , whose representations are the often used spinor fields , is usually constructed as a subset of the tensor fields with values ​​in the Clifford algebra .