Differential geometry MOC

Tensor field

Let be a -manifold. A -tensor field is a generalization of a vector field where we assign a tensor smoothly to every point in . A homogenous tensor field of type is a -multilinear map

where and denote the spaces of 1-forms and vector fields respectively. The -module of all such tensor fields is denoted . A general nonhomogenous tensor field is a direct sum of tensor fields.

As a section

The above definition is equivalent to a -section of the tensor product of copies of the tangent bundle and copies of the cotangent bundle

and a general (non-homogenous) tensor field is a -section of a sum bundle.

Proof

proof

Further terminology

• A totally contravariant, totally antisymmetric tensor field is a Differential form.
• Often physicists view the Tensor field transformation laws as the definition of a tensor field.
• A metric tensor enables Raising and lowering of indices.

Local coördinates

Let be a chart. Restricted to , we may write a smooth tensor field in the form

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