Tensor on a vector space

over a field

An element of the vector space

where is the dual space of . The tensor is said to be times contravariant and times covariant, or to be of type . The number is called the contravariant valency, and the covariant valency, while the number is called the general valency of the tensor . The space is identified with . Tensors of type are called contravariant, those of the type are called covariant, and the remaining ones are called mixed.

Examples of tensors.

1) A vector of the space (a tensor of type ).

2) A covector of the space (a tensor of type ).

3) Any covariant tensor

where , defines a -linear form on by the formula

the mapping from the space into the space of all -linear forms on is linear and injective; if , then this mapping is an isomorphism, since any -linear form corresponds to some tensor of type .

4) Similarly, a contravariant tensor in defines a -linear form on , and if is finite dimensional, the converse is also true.

5) Every tensor

where and , defines a linear transformation of the space given by the formula

if , any linear transformation of the space is defined by a tensor of type .

6) Similarly, any tensor of type defines in a bilinear operation, that is, the structure of a -algebra. Moreover, if , then any -algebra structure in is defined by a tensor of type , called the structure tensor of the algebra.

Let be finite dimensional, let be a basis of it, and let be the dual basis of the space . Then the tensors

form a basis of the space . The components of a tensor with respect to this basis are also called the components of the tensor with respect to the basis of the space . For instance, the components of a vector and of a covector coincide with their usual coordinates with respect to the bases and ; the components of a tensor of type coincide with the entries of the matrix corresponding to the bilinear form; the components of a tensor of type coincide with the entries of the matrix of the corresponding linear transformation, and the components of the structure tensor of an algebra coincide with its structure constants. If is another basis of , with , and , then the components of the tensor in this basis are defined by the formula

(1)

Here, as often happens in tensor calculus, Einstein's summation convention is applied: with respect to any pair of equal indices of which one is an upper index and the other is a lower index, it is understood that summation from 1 to is carried out. Conversely, if a system of elements of a field depending on the basis of the space is altered in the transition from one basis to another basis according to the formulas (1), then this system is the set of components of some tensor of type .

In the vector space the operations of addition of tensors and of multiplication of a tensor by a scalar from are defined. Under these operations the corresponding components are added, or multiplied by the scalar. The operation of multiplying tensors of different types is also defined; it is introduced as follows. There is a natural isomorphism of vector spaces

mapping

to

Consequently, for any and the element can be regarded as a tensor of type and is called the tensor product of and . The components of the product are computed according to the formula

Let , , and let the numbers and be fixed with and . Then there is a well-defined mapping such that

It is called contraction in the -th contravariant and the -th covariant indices. In components, the contraction is written in the form

For instance, the contraction of a tensor of type is the trace of the corresponding linear transformation.

A tensor is similarly defined on an arbitrary unitary module over an associative commutative ring with a unit. The stated examples and properties of tensors are transferred, with corresponding changes, to this case, it being sometimes necessary to assume that is a free or a finitely-generated free module.

Let a non-degenerate bilinear form be fixed in a finite-dimensional vector space over a field (for example, is a Euclidean or pseudo-Euclidean space over ); in this case the form is called a metric tensor. A metric tensor defines an isomorphism by the formula

Let , and let the index , , be fixed. Then the formula

defines an isomorphism , called lowering of the -th contravariant index. In other terms,

In components, lowering an index has the form

Similarly one defines the isomorphism of raising the -th covariant index :

which maps onto . In components, raising an index is written in the form

where . In particular, raising at first the first, and then also the remaining covariant index of the metric tensor leads to a tensor of type with components (a contravariant metric tensor). Sometimes the lowered (raised) index is not moved to the first (last) place, but is written in the same place in the lower (upper) group of indices, a point being put in the empty place which arises. For instance, for the components of the tensor are written in the form .

Any linear mapping of vector spaces over defines in a natural way linear mappings

and

If is an isomorphism, the linear mapping

is also defined and . The correspondence has functorial properties. In particular, it defines a linear representation of the group in the space (the tensor representation).

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2]	I.M. Gel'fand, "Lectures on linear algebra" , Interscience (1961) (Translated from Russian)
[3]	A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian)
[4]	M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian)
[5]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)