# Tensor taper

The tensor taper or contraction is a mathematical term from linear algebra . It is a generalization of the trace of a linear mapping to tensors that are at least singly covariant and singly contravariant.

## definition

Let be a finite-dimensional vector space and be ${\ displaystyle V}$

${\ displaystyle T_ {s} ^ {r} (V): = \ underbrace {V \ otimes \ cdots \ otimes V} _ {r {\ text {-mal}}} \ otimes \ underbrace {V ^ {*} \ otimes \ cdots \ otimes V ^ {*}} _ {s {\ text {times}}}}$

the tensor space of -fold contravariant and -fold covariant tensors (short: -Tensors) over . ${\ displaystyle r}$${\ displaystyle s}$${\ displaystyle (r, s)}$${\ displaystyle V}$

The linear mapping is called the tapering or contraction of a tensor (more precisely: contraction) ${\ displaystyle (k, l)}$

${\ displaystyle C_ {l} ^ {k}: T_ {s} ^ {r} (V) \ rightarrow T_ {s-1} ^ {r-1} (V)}$

with and , which by ${\ displaystyle 1 \ leq k \ leq r}$${\ displaystyle 1 \ leq l \ leq s}$

${\ displaystyle v_ {1} \ otimes \ cdots \ otimes v_ {r} \ otimes \ xi _ {1} \ otimes \ cdots \ otimes \ xi _ {s} \ mapsto}$
${\ displaystyle \ xi _ {l} (v_ {k}) (v_ {1} \ otimes \ cdots \ otimes v_ {k-1} \ otimes v_ {k + 1} \ otimes \ cdots \ otimes v_ {r} \ otimes \ xi _ {1} \ otimes \ cdots \ otimes \ xi _ {l-1} \ otimes \ xi _ {l + 1} \ otimes \ cdots \ otimes \ xi _ {s})}$

can be defined. There is an element of . Not every element of is of this form, but the elements of this form create the tensor space and the mapping is well defined. If one sets , then a tensor -th level becomes a tensor of level . ${\ displaystyle v_ {1} \ otimes \ cdots \ otimes v_ {r} \ otimes \ xi _ {1} \ otimes \ cdots \ otimes \ xi _ {s}}$${\ displaystyle T_ {s} ^ {r} (V)}$${\ displaystyle T_ {s} ^ {r} (V)}$${\ displaystyle n: = r + s}$${\ displaystyle n}$${\ displaystyle n-2}$

## Examples

• If one interprets a matrix as a simply co- and contravariant tensor, the taper of a matrix is ​​its trace. This can be seen very quickly if the matrix is ​​represented as a linear combination . Here one forms the basis of and that forms the dual basis of . If one now applies the function , one obtains. This shows that the tensor taper is a generalization of the trace operator known from linear algebra . For this reason, the mapping is also called track formation.${\ displaystyle A \ in \ operatorname {End} (V) \ cong V \ otimes V ^ {*}}$
${\ displaystyle A = \ sum _ {i, j} \ lambda _ {i} ^ {j} \, v_ {i} \ otimes \ xi _ {j}}$
${\ displaystyle v_ {i}}$${\ displaystyle V}$${\ displaystyle \ xi _ {j}}$${\ displaystyle V ^ {*}}$${\ displaystyle C_ {1} ^ {1}}$
${\ displaystyle C_ {1} ^ {1} (A) = C_ {1} ^ {1} (\ sum _ {i, j} \ lambda _ {i} ^ {j} \, v_ {i} \ otimes \ xi _ {j}) = \ sum _ {i, j} \ lambda _ {i} ^ {j} \ delta _ {ij} = \ sum _ {i} \ lambda _ {i} ^ {i} = \ operatorname {track} (A).}$
• The Ricci tensor is obtained from the Riemann curvature tensor by tapering .${\ displaystyle R_ {ijk} ^ {l}}$ ${\ displaystyle R_ {ik} = R_ {ijk} ^ {j}}$

## literature

• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York NY et al. 1988, ISBN 0-387-96790-7 .

## Individual evidence

1. Ulrich E. Schröder: Special Theory of Relativity . German, Frankfurt am Main 2005, ISBN 3-8171-1724-8 , pp. 51 .