In physics, contraction is a dilation with coefficient . This notion is used in fixed point theory, theory of topological vector spaces etc. There is also a notion of contraction from metric space theory; see short map. Finally, the contraction rule is a structural rule in logic and type theory. This entry will be predominantly about another notion of a contraction.
This entry will be predominantly about contraction of tensors, where by tensor we mean a vector in some tensor power of a vector -space (or a projective -module if is only a commutative ring). Let be the dual vector space and be some tensor of . Then one may define -contraction
by pairing by the evaluation map the -th tensor factor of and -th tensor factor of . In fact as a map written, one can contract also elements of which did not come from a product of a pair of element (i.e. which are not decomposable tensors).
Let the rank of be finite. If is given in some basis by components and is given in the dual basis by components , then the components of the contraction will be
More generally one can contract mixed tensors and do several contractions simultaneously. In fact it is better to think of a contraction as a tensor multiplication of two tensors and then doing the genuine contraction of one upper and one lower index of the same tensor:
The simplest case is the trace of a -tensor: .
These operations can be symmetrized or antisymmetrised appropriately to make sense on symmetric or antisymmetric powers.
For example, there is a contraction of a vector and a -form :
and is a graded derivation of the exterior algebra of degree . This is also done for the tangent bundle which is a -module , then one gets the contraction of vector fields and differential forms. It can also be done in vector spaces, fibrewise.
wikipedia: tensor contraction
Shlomo Sternberg, Introduction to differential geometry