Tensor Fields

Tensors are mathematical objects that generalize the familiar concepts of scalars, vectors, and exterior forms to higher dimensions. Their crucial feature lies in their ability to represent physical laws in a form that remains valid regardless of the coordinate system used by an observer.

Covectors and The Cotangent Bundle

Covector & Cotangent Space

A covector at is an element in the dual of the tangent space, i.e. a linear map from to . The dual of , which we denote , and is called the cotangent space at .

Similar to the tangent space, the set of all covectors on forms a vector bundle over , denoted , called the cotangent bundle:

Cotangent Bundle

A cotangent bundle is the disjoint union of the cotangent spaces of a manifold :

Tensor and Tensor Fields

Tensor (Field) on Manifold

A tensor of type at is a -tensor over . A tensor field of type on is a smooth assignment of a tensor of type at each point . Inherited from the terminology of tensors, tensor fields of type is said to be covariant, and tensor fields of type is said to be contravariant. The set of all tensors of type at forms a vector space, denoted , and the set of all tensors of type on forms a vector bundle over , denoted .

Remark

One can naturally recognize every tensor field as a map .

e.g.

• A covector is just a tensor of type , and a vector is a tensor of type .
• Real-valued functions are covariant, vector fields are contravariant.
• One-forms or covectors are also covariant. This is nice, because the prefix “co-” means not only duality, but also covariance.

Tensors in Local Coordinates

Tensor Components

Let be a coordinate on . If is a tensor field, the components of relative to this coordinate are the real valued functions: where all indices run from to .

Contraction Tensor

Let be a tensor at of type . Then there is a tensor of type , the contraction of with respect to the -th covector and the -th vector is a tensor at of type given by

e.g. A special case is where is a tensor of type . This takes a vector and gives another vector, and so is nothing other than a linear operator on . There is a unique contraction in this case, which is just the trace of the operator:

Raising and Lowering Indices

Lowering and raising indices is the procedure of converting between covariant (lower) and contravariant (upper) components of tensors by contracting with the metric tensor or its inverse .