Secondary Vector Bundle Structure and Canonical Flip
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 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.
Read more about this topic: Double Tangent Bundle
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