Natural bundle

In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle ${\displaystyle F^{s}(M)}$ for some ${\displaystyle s\geq 1}$. It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold ${\displaystyle M}$ together with their partial derivatives up to order at most ${\displaystyle s}$.[1]

The concept of a natural bundle was introduced by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.[2]

An example of natural bundle (of first order) is the tangent bundle ${\displaystyle TM}$ of a manifold ${\displaystyle M}$.