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