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Tensor transition laws define a smooth vector bundle

Statement

For every smooth manifold $M$ and integers $r, s \geq 0$ , the tensor-coordinate change rules define a smooth vector bundle whose fibre over $p$ is the space of type $(r, s)$ tensors on $T_{p} M$ .

Facts & Assumptions

Given: A smooth manifold $M$ with overlapping charts $(U, x)$ and $(V, y)$ .

[F1]

The fibre of $T_{s}^{r} M$ at $p$ is the space of type $(r, s)$ tensors on $T_{p} M$ (The type $(r, s)$ tensor bundle).

[L1]

Tangent bases transform by the Jacobian, and cotangent bases transform by the inverse transpose Jacobian (Change-of-coordinate formula for tangent bases, Cotangent coordinate changes use the inverse transpose Jacobian).

[L2]

A smooth cocycle of fibrewise linear transition maps defines a smooth vector bundle (Construction of a vector bundle from a smooth cocycle).

Proof

technique · direct

1.1F1givenconstruct

On a chart domain $U$ , the coordinate bases $\partial / \partial x^{i}$ and [F1, given, construct] $d x^{i}$ identify each fibre in [F1] with the fixed finite-dimensional vector space $Mult (((R^{m})^{*})^{r} \times (R^{m})^{s}, R)$ of type $(r, s)$ tensors on $R^{m}$ . This gives local trivializations $T_{s}^{r} M ∣_{U} ≅ U \times Mult (((R^{m})^{*})^{r} \times (R^{m})^{s}, R) .$

2.1L1step 1.1algebra

On an overlap, [L1] shows that each contravariant slot picks up one inverse [L1, step 1.1, algebra] Jacobian factor and each covariant slot picks up one Jacobian factor. Hence the tensor-coordinate change map is fibrewise linear, smooth in the base point, and satisfies the cocycle law because Jacobians and inverse Jacobians do.

3.1F1L2step 2.1

Therefore [L2] applies to these local transition maps and produces a smooth [F1, L2, step 2.1] vector bundle. By construction its fibre over $p$ is the tensor space from [F1].

4.1step 3.1∎

Thus the tensor transition laws define the smooth tensor bundle $T_{s}^{r} M$ . [step 3.1]

Tensor transition laws define a smooth vector bundle

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources