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Concept
Double tangent bundle
Summary
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 in 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
The canonical flip has the property that for any f: R2 → M,
where s and t are coordinates of the standard basis of R 2. Note that both partial derivatives are functions from R2 to TTM.
This property can, in fact, be used to give an intrinsic definition of the canonical flip. Indeed, there is a submersion
p: J20 (R2,M) → TTM given by
where p can be defined in the space of two-jets at zero because only depends on f up to order two at zero. We consider the application:
where α(s,t)= (t,s). Then J is compatible with the projection p and induces the canonical flip on the quotient TTM.
As for any vector bundle, the tangent spaces Tξ(TxM) of the fibres TxM of the tangent bundle (TM,πTM,M) can be identified with the fibres TxM themselves.
Official source
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