Natural bundle

A bundle $E \to M$ is natural if any diffeomorphism $F : M \to M$ induces a canonical lift $\hat{F} : E \to E$ covering $F$ , in a way that is functorial (i.e., compatible with composition and identities).

Some examples of natural bundles include:

The Tangent Bundle ( $T M \to M$ ): For a diffeomorphism $F : M \to M$ , the canonical lift $\hat{F} : T M \to T M$ is simply the pushforward (or differential) $F_{*} : T M \to T M$ . This maps a tangent vector $v \in T_{p} M$ to $F_{*} (v) \in T_{F (p)} M$ .
Tensor Bundles ( $T_{l}^{k} M \to M$ ): More broadly, any tensor bundle, such as the bundle of $(k, l)$ -tensors $T_{l}^{k} M = ⨂^{k} T M \otimes ⨂^{l} T^{*} M$ , is natural. A diffeomorphism $F : M \to M$ lifts by applying the pushforward $F_{*}$ to tangent vector components and the pullback $(F^{- 1})^{*}$ to cotangent vector components.
Lorentzian Metric Bundles: This is a specific type of tensor bundle. A Lorentzian metric at a point is a symmetric, non-degenerate $(0, 2)$ -tensor with a specific signature (e.g., $(1, n - 1)$ ). The bundle of Lorentzian metrics is an open subbundle of the more general bundle of $(0, 2)$ -tensors ( $T_{2}^{0} M \to M$ ), consisting only of those tensors that meet the symmetry, non-degeneracy, and signature requirements. The natural lift is given by the pullback of the metric. This is related to the "general covariance" notion in GR. See on theories, symmetries and gauge.