What Is the Sacral Plexus?

(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 ^[2] .

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 ^[2] .

(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