Natural bundle

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

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.

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