tensor product of vector bundles E, F (over the same space X) is a vector bundle, denoted by E ⊗ F, whose fiber over each point x ∈ X is the tensor product...

v^{k}} transforms by the inverse Jacobian. A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space...

mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold...

section of the tensor product bundle E* ⊗ E*. The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called...

example, one can define a tensor field on a smooth manifold M as a (global or local) section of the tensor product (called tensor bundle) ( T M ) ⊗ p ⊗ O ( T...

F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum Ex ⊕ Fx of the vector spaces Ex and Fx. The tensor product bundle E ⊗ F is defined...

v\otimes w} is called the tensor product of v and w. An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of two vectors is sometimes...

the dual vector bundle E ∗ {\displaystyle E^{*}} , tensor powers E ⊗ k {\displaystyle E^{\otimes k}} , symmetric and antisymmetric tensor powers S k E ...

differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing...

(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), and general relativity (stress–energy tensor, curvature tensor, ...). In...

differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors...

product. The Hom bundle H o m ( E 1 , E 2 ) {\displaystyle \mathrm {Hom} (E_{1},E_{2})} of two vector bundles is canonically isomorphic to the tensor...

vector bundle endowed with a bundle metric and its dual. Given a (0, 2) tensor X = Xij ei ⊗ ej, we define the trace of X through the metric tensor g by...

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...

of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or...

{\displaystyle n} and a tensor of order m {\displaystyle m} is a tensor of order n + m − 2 {\displaystyle n+m-2} , see Tensor contraction for details...

relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of...

(also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors of W in the product). Then the pullback...

the product is not a field) "Categorified" concepts, applied "pointwise" on objects and morphisms: Tensor product of vector bundles Tensor product of sheaves...

{\displaystyle (r,s,r',s')} is defined as a section of the exterior tensor product bundle T s r M ⊠ T s ′ r ′ M {\displaystyle T_{s}^{r}M\boxtimes T_{s'}^{r'}M}...

mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear...

be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T∗M (see pseudotensor)....

global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by tensor product. The same construction...

line bundle L is big if and only if it has a positive tensor power which is the tensor product of an ample line bundle A and an effective line bundle B (meaning...

of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory...

tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product....

Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann...

strength tensor is a rank 2 tensor field on the spacetime with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see vector bundle for...

p is a smooth section of the tensor product bundle of E with Λp(T ∗M), the p-th exterior power of the cotangent bundle of M. The space of such forms...

consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to...