Double Tangent Bundle - Secondary Vector Bundle Structure and Canonical Flip

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

\xi = \xi^k\frac{\partial}{\partial x^k}\Big|_x\in T_xM, \qquad X = X^k\frac{\partial}{\partial x^k}\Big|_x\in T_xM

and apply the associated coordinate system

\xi \mapsto (x^1,\ldots,x^n,\xi^1,\ldots,\xi^n)

on TM. Then the fibre of the secondary vector bundle structure at X∈TxM takes the form

(\pi_{TM})^{-1}_*(X) = \Big\{ \ X^k\frac{\partial}{\partial x^k}\Big|_\xi + Y^k\frac{\partial}{\partial\xi^k}\Big|_\xi \ \Big| \ \xi\in T_xM \, \ Y^1,\ldots,Y^n\in\R \ \Big\}.

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

j\Big(X^k\frac{\partial}{\partial x^k}\Big|_\xi + Y^k\frac{\partial}{\partial \xi^k}\Big|_\xi\Big) = \xi^k\frac{\partial}{\partial x^k}\Big|_X + Y^k\frac{\partial}{\partial \xi^k}\Big|_X.

The canonical flip has the property that for any f: R2 → M,

\frac {\partial f} {{\partial t} {\partial s}} = j \circ \frac {\partial f} {{\partial s} {\partial t}}

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

p=\frac {\partial f} {{\partial t} {\partial s}} (0,0)

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:

J: J^2_0(\mathbb{R}^2,M) \to J^2_0(\mathbb{R}^2,M) \quad / \quad J=

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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