(r, s) tensor bundle of type (r, s) is a generalization of the concepts of tangent bundle and cotangent bundle. The so-called (r, s) -type tensor bundles refer to the non-intersection of (r, s) -type tensor spaces in the tangent space at points on the differential manifold M, that is, the (r, s) -type tensor bundles T
r, s on M (M) =
pM (Tp (M))
r s, where (T
p (M))
r s represents a (r, s) -type tensor space of T
p M.
(r, s) tensor bundle of type (r, s) is a generalization of the concepts of tangent bundle and cotangent bundle. The term (r, s) tensor bundle refers to the non-intersection of the (r, s) tensor space of the tangent space at each point on the differential manifold M, that is, the (r, s) tensor bundle on M
among them Express (R, s) -type tensor space.
(1,0) tensor bundles are tangent bundles, and (0,1) tensor bundles are
(r, s) -type tensor bundle projection
Let M be n-dimensional Manifold, with It is the tangent space and cotangent space of M at a, respectively, so each point a of manifold M has (r, s) -type tensor space.
this is Dimensional vector space .
make in Introduced topology on Hausdorff space with countable basis, said Is a (r, s) tensor bundle on manifold M.
Consider a coordinate system of manifold M At any point with Natural bases with therefore, Base
And for any Have
among them .
Definition Is an (r, s) tensor bundle on M, mapping ,defined as Have Weigh It is the natural projection on the tensor bundle, referred to as bundle projection for short.
Assume Yes Map if
(Identity mapping on M)
For any Then said Tensor bundle A smooth section, or (r, s) type smooth tensor field on M.
Cut bundle The section is the tangent vector field of M, and the cotangent bundle The cross section of is the differential 1-form on M .
(r, s) -type tensor bundles, cotangent bundles
In (r, s) tensor bundle In We get the (1,0) -type tensor bundle tangent bundle on the manifold M, that is, the tangent bundle is expressed as
If We get a cotangent bundle of type (0,1) tensor bundle on manifold M, that is, the cotangent bundle is expressed as
