Double Tangent Bundle - Secondary Vector Bundle Structure and Canonical Flip | Secondary Vector Bundle Structure Canonical Flip

Secondary Vector Bundle Structure and Canonical Flip

Since (TM,π_TM,M) is a vector bundle on its own right, its tangent bundle has the secondary vector bundle structure (TTM,(π_TM)_*,TM), where (π_TM)_*:TTM→TM is the push-forward of the canonical projection π_TM: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∈T_xM 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,π_TTM,TM) and (TTM,(π_TM)_*,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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