Double tangent bundle
In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M . A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM.
The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle.
Since (TM,πTM,M) is a vector bundle on its own right, its tangent bundle has the secondary vector bundle structure (TTM,(πTM)*,TM), where (πTM)*:TTM→TM is the push-forward of the canonical projection πTM:TM→M. In the following we denote
and apply the associated coordinate system
on TM. Then the fibre of the secondary vector bundle structure at X∈TxM takes the form
The double tangent bundle is a double vector bundle.
The canonical flip is a smooth involution j:TTM→TTM that exchanges these vector space structures in the sense that it is a vector bundle isomorphism between (TTM,πTTM,TM) and (TTM,(πTM)*,TM). In the associated coordinates on TM it reads as
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