In physics, contraction is a dilation with coefficient $\lambda\lt 1$. 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 $V^{\otimes n}$ of a vector $k$-space $V$ (or a projective $k$-module if $k$ is only a commutative ring). Let $V^* = Hom_k(V,k)$ be the dual vector space and $(V^*)^{\otimes m}$ be some tensor of $V^*$. Then one may define $(l,s)$-contraction
by pairing by the evaluation map the $l$-th tensor factor of $(V^*)^{\otimes r}$ and $s$-th tensor factor of $V^{\otimes n}$. In fact as a map written, one can contract also elements of $(V^*)^{\otimes m}\otimes V^{\otimes n}$ which did not come from a product of a pair of element (i.e. which are not decomposable tensors).
Let the rank $r$ of $V$ be finite. If $S\in V^{\otimes n}$ is given in some basis by components $S^{i_1,\ldots, i_n}$ and $T\in (V^*)^{\otimes r}$ is given in the dual basis by components $T_{j_1,\ldots,j_r}$, 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 $(1,1)$-tensor: $tr A = \sum_{i=1}^r A^i_i$.
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 $X\in V$ and a $n$-form $\omega\in \Lambda V^*$:
and $\iota_X: \omega\mapsto \iota_X(\omega)$ is a graded derivation of the exterior algebra of degree $-1$. This is also done for the tangent bundle which is a $C^\infty(M)$-module $V = T M$, 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
Last revised on March 28, 2021 at 00:20:58. See the history of this page for a list of all contributions to it.