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| In [[mathematics]], particularly [[differential topology]], the '''double tangent bundle''' or the '''second tangent bundle''' refers to the [[tangent bundle]] {{nowrap|(''TTM'',''π''<sub>''TTM''</sub>,''TM'')}} of the total space ''TM'' of the [[tangent bundle]] {{nowrap|(''TM'',''π''<sub>''TM''</sub>,''M'')}} of a [[differentiable manifold|smooth manifold]] ''M''
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| .<ref>J.M.Lee, ''Introduction to Smooth Manifolds'', Springer-Verlag, 2003.</ref> A note on notation: in this article, we denote projection maps by their domains, e.g., ''π''<sub>''TTM''</sub> : ''TTM'' → ''TM''. Some authors index these maps by their ranges instead, so for them, that map would be written ''π''<sub>''TM''</sub>.
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| The second tangent bundle arises in the study of [[Connection (vector bundle)|connections]] and second order ordinary differential equations, i.e., [[spray (mathematics)|(semi)spray structures]] on smooth manifolds, and it is not to be confused with the [[jet bundle|second order jet bundle]].
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| == Secondary vector bundle structure and canonical flip ==
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| Since {{nowrap|(''TM'',''π''<sub>''TM''</sub>,''M'')}} is a vector bundle on its own right, its tangent bundle has the [[secondary vector bundle structure]] {{nowrap|(''TTM'',(''π''<sub>''TM''</sub>)<sub>*</sub>,''TM''),}} where {{nowrap|(''π''<sub>''TM''</sub>)<sub>*</sub>:''TTM''→''TM''}} is the push-forward of the canonical projection {{nowrap|''π''<sub>''TM''</sub>:''TM''→''M''.}}
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| In the following we denote
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| :<math> | |
| \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
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| </math>
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| and apply the associated coordinate system
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| :<math>
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| \xi \mapsto (x^1,\ldots,x^n,\xi^1,\ldots,\xi^n)
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| </math>
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| on ''TM''. Then the fibre of the secondary vector bundle structure at ''X''∈''T''<sub>''x''</sub>''M'' takes the form
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| :<math>
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| (\pi_{TM})^{-1}_*(X) = \Big\{ \ X^k\frac{\partial}{\partial x^k}\Big|_\xi + Y^k\frac{\partial}{\partial\xi^k}\Big|_\xi
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| \ \Big| \ \xi\in T_xM \ , \ Y^1,\ldots,Y^n\in\R \ \Big\}.
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| </math>
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| The '''canonical flip'''<ref>P.Michor. ''Topics in Differential Geometry,'' American Mathematical Society, 2008.</ref> is a smooth involution ''j'':''TTM''→''TTM'' that exchanges these vector space structures
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| in the sense that it is a vector bundle isomorphism between {{nowrap|(''TTM'',''π''<sub>''TTM''</sub>,''TM'')}} and {{nowrap|(''TTM'',(''π''<sub>''TM''</sub>)<sub>*</sub>,''TM'').}} In the associated coordinates on ''TM'' it reads as
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| :<math>
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| j\Big(X^k\frac{\partial}{\partial x^k}\Big|_\xi + Y^k\frac{\partial}{\partial \xi^k}\Big|_\xi\Big)
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| = \xi^k\frac{\partial}{\partial x^k}\Big|_X + Y^k\frac{\partial}{\partial \xi^k}\Big|_X. | |
| </math>
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| The canonical flip has the property that for any ''f'': '''R'''<sup>2</sup> → ''M'',
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| :<math>
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| \frac {\partial f} {{\partial t} {\partial s}} = j \circ \frac {\partial f} {{\partial s} {\partial t}}
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| </math>
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| where ''s'' and ''t'' are coordinates of the standard basis of '''R''' <sup>2</sup>. Note that both partial derivatives are functions from '''R'''<sup>2</sup> to ''TTM''.
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| This property can, in fact, be used to give an intrinsic definition of the canonical flip.<ref>Robert J. Fisher and H. Turner Laquer, Second Order Tangent Vectors in Riemannian Geometry, J. Korean Math. Soc. 36 (1999), No. 5, pp. 959-1008</ref> Indeed, there is a submersion
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| ''p'': J<sup>2</sup><sub>0</sub> ('''R'''<sup>2</sup>,M) → ''TTM'' given by
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| :<math>
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| p([f])=\frac {\partial f} {{\partial t} {\partial s}} (0,0)
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| </math>
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| 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:
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| :<math>
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| J: J^2_0(\mathbb{R}^2,M) \to J^2_0(\mathbb{R}^2,M) \quad / \quad J([f])=[f \circ \alpha]
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| </math>
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| where α(''s'',''t'')= (''t'',''s''). Then ''J'' is compatible with the projection ''p'' and induces the canonical flip on the quotient ''TTM''.
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| == Canonical tensor fields on the tangent bundle==
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| As for any [[vector bundle]], the tangent spaces {{nowrap|''T''<sub>ξ</sub>(''T''<sub>''x''</sub>''M'')}} of the fibres ''T''<sub>''x''</sub>''M'' of the tangent bundle {{nowrap|(''TM'',''π''<sub>''TM''</sub>,''M'')}} can be identified with the fibres ''T''<sub>x</sub>''M'' themselves. Formally this is achieved though the '''vertical lift''', which is a natural vector space isomorphism
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| {{nowrap|vl<sub>ξ</sub>:''T''<sub>''x''</sub>''M''→''V''<sub>ξ</sub>(''T''<sub>''x''</sub>''M'')}} defined as
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| :<math>
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| (\operatorname{vl}_\xi X)[f]:=\frac{d}{dt}\Big|_{t=0}f(x,\xi+tX), \qquad f\in C^\infty(TM).
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| </math>
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| The vertical lift can also be seen as a natural vector bundle isomorphism
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| {{nowrap|vl:(π<sub>''TM''</sub>)<sup>*</sup>''TM''→''VTM''}}
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| from the pullback bundle of {{nowrap|(''TM'',''π''<sub>''TM''</sub>,''M'')}} over {{nowrap|''π''<sub>''TM''</sub>:''TM''→''M''}} onto the vertical tangent bundle
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| :<math>
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| VTM:=\operatorname{Ker}(\pi_{TM})_* \subset TTM.
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| </math>
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| The vertical lift lets us define the '''canonical vector field'''
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| :<math>
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| V:TM\to TTM; \qquad V_\xi := \operatorname{vl}_\xi\xi,
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| </math>
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| which is smooth in the slit tangent bundle ''TM''\0. The canonical vector field can be also defined as the infinitesimal generator of the Lie-group action
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| :<math>
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| \mathbb R\times (TM\setminus 0) \to TM\setminus 0; \qquad (t,\xi) \mapsto e^t\xi.
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| </math>
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| Unlike the canonical vector field, which can be defined for any vector bundle, the '''canonical endomorphism'''
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| :<math>
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| J:TTM\to TTM; \qquad J_\xi X := \operatorname{vl}_\xi(\pi_{TM})_*X, \qquad X\in T_\xi TM
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| </math>
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| is special to the tangent bundle. The canonical endomorphism ''J'' satisfies
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| :<math>
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| \operatorname{Ran}(J)=\operatorname{Ker}(J)=VTM, \qquad \mathcal L_VJ= -J, \qquad J[X,Y]=J[JX,Y]+J[X,JY],
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| </math>
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| and it is also known as the '''tangent structure''' for the following reason. If (''E'',''p'',''M'') is any vector bundle
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| with the canonical vector field ''V'' and a (1,1)-tensor field ''J'' that satisfies the properties listed above, with ''VE'' in place of ''VTM'', then the vector bundle (''E'',''p'',''M'') is isomorphic to the tangent bundle {{nowrap|(''TM'',''π''<sub>''TM''</sub>,''M'')}} of the base manifold, and ''J'' corresponds to the tangent structure of ''TM'' in this isomorphism.
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| There is also a stronger result of this kind <ref>D.S.Goel, ''Almost Tangent Structures'', Kodai Math.Sem.Rep. '''26''' (1975), 187-193.</ref> which states that if ''N'' is a 2''n''-dimensional manifold and if there exists a (1,1)-tensor field ''J'' on ''N'' that satisfies
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| :<math>
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| \operatorname{Ran}(J)=\operatorname{Ker}(J), \qquad J[X,Y]=J[JX,Y]+J[X,JY],
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| </math>
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| then ''N'' is diffeomorphic to an open set of the total space of a tangent bundle of some ''n''-dimensional manifold ''M'', and ''J'' corresponds to the tangent structure of ''TM'' in this diffeomorphism.
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| In any associated coordinate system on ''TM'' the canonical vector field and the canonical endomorphism have the coordinate representations
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| :<math>
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| V = \xi^k\frac{\partial}{\partial \xi^k}, \qquad J = dx^k\otimes\frac{\partial}{\partial \xi^k}.
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| </math>
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| == (Semi)spray structures ==
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| A [[Spray (mathematics)|Semispray structure]] on a smooth manifold ''M'' is by definition a smooth vector field ''H'' on ''TM'' \0 such that ''JH''=''V''. An equivalent definition is that ''j''(''H'')=''H'', where ''j'':''TTM''→''TTM'' is the canonical flip. A semispray ''H'' is a [[Spray (mathematics)|spray]], if in addition, [''V'',''H'']=''H''.
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| Spray and semispray structures are invariant versions of second order ordinary differential equations on ''M''. The difference between spray and semispray structures is that the solution curves of sprays are invariant in positive reparametrizations as point sets on ''M'', whereas solution curves of semisprays typically are not.
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| == Nonlinear covariant derivatives on smooth manifolds ==
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| The canonical flip makes it possible to define nonlinear covariant derivatives on smooth manifolds as follows. Let
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| :<math>
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| T(TM\setminus 0) = H(TM\setminus 0) \oplus V(TM\setminus 0)
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| </math>
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| be an [[Ehresmann connection]] on the slit tangent bundle ''TM''/0 and consider the mapping
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| :<math>
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| D:(TM\setminus 0)\times \Gamma(TM) \to TM; \quad D_XY := (\kappa\circ j)(Y_*X),
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| </math>
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| where ''Y''<sub>*</sub>:''TM''→''TTM'' is the push-forward, ''j'':''TTM''→''TTM'' is the canonical flip and κ:''T''(''TM''/0)→''TM''/0 is the connector map. The mapping ''D''<sub>''X''</sub> is a derivation in the module Γ (''TM'') of smooth vector fields on ''M'' in the sense that
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| * <math>D_X(\alpha Y + \beta Z) = \alpha D_XY + \beta D_XZ, \qquad \alpha,\beta\in\mathbb R</math>.
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| * <math>D_X(fY) = X[f]Y + f D_XY, \qquad \qquad \qquad f\in C^\infty(M)</math>.
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| Any mapping ''D''<sub>''X''</sub> with these properties is called a '''(nonlinear) covariant derivative'''
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| <ref>I.Bucataru, R.Miron, ''Finsler-Lagrange Geometry'', Editura Academiei Române, 2007.</ref> on ''M''.
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| The term ''nonlinear'' refers to the fact that this kind of covariant derivative ''D''<sub>''X''</sub> on is not necessarily linear with respect to the direction ''X''∈''TM''/0 of the differentiation.
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| Looking at the local representations one can confirm that the Ehresmann connections on (''TM''/0,π<sub>''TM''/0</sub>,''M'') and nonlinear covariant derivatives on ''M'' are in one-to-one correspondence. Furthermore, if ''D''<sub>''X''</sub> is linear in ''X'', then the Ehresmann connection is linear in the [[secondary vector bundle structure]], and ''D''<sub>''X''</sub> coincides with its linear covariant derivative if and only if the torsion of the connection vanishes.
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| == See also ==
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| * [[Spray (mathematics)]]
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| * [[Secondary vector bundle structure]]
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| * [[Finsler manifold]]
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| == References ==
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| <references />
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| [[Category:Differential geometry]]
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| [[Category:Topology]]
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The Google Campaign team�aforementioned today that it is pealing out�an activeness swarm to assist you reminder changes to your files and folders. A new ?? release that appears at the go past flop turning point of the screen door gives you access code to the stream, which notes actions interpreted on files and folders in your Drive, and WHO took them. Edits, comments, young files, changed Indian file names and more than volition all at present seem within the flow. The thought is naught new�?
? Dropbox and Loge accept offered standardised activeness streams for quite close to clock ?? just it testament expected be receive news program to anyone who collaborates using Labor.
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