Double tangent bundle

In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,π_TTM,TM) of the total space TM of the tangent bundle (TM,π_TM,M) of a smooth manifold M .^[1] A note on notation: in this article, we denote projection maps by their domains, e.g., π_TTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written π_TM.

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.

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 double tangent bundle is a double vector bundle.

The canonical flip^[2] 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.^[3] Indeed, there is a submersion p: J²₀ (R²,M) → TTM given by

p([f])=\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([f])=[f \circ \alpha]

where α(s,t)= (t,s). Then J is compatible with the projection p and induces the canonical flip on the quotient TTM.

Canonical tensor fields on the tangent bundle

As for any vector bundle, the tangent spaces T_ξ(T_xM) of the fibres T_xM of the tangent bundle (TM,π_TM,M) can be identified with the fibres T_xM themselves. Formally this is achieved though the vertical lift, which is a natural vector space isomorphism vl_ξ:T_xM→V_ξ(T_xM) defined as

(\operatorname{vl}_\xi X)[f]:=\frac{d}{dt}\Big|_{t=0}f(x,\xi+tX), \qquad f\in C^\infty(TM).

The vertical lift can also be seen as a natural vector bundle isomorphism vl:(π_TM)^*TM→VTM from the pullback bundle of (TM,π_TM,M) over π_TM:TM→M onto the vertical tangent bundle

VTM:=\operatorname{Ker}(\pi_{TM})_* \subset TTM.

The vertical lift lets us define the canonical vector field

V:TM\to TTM; \qquad V_\xi := \operatorname{vl}_\xi\xi,

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

\mathbb R\times (TM\setminus 0) \to TM\setminus 0; \qquad (t,\xi) \mapsto e^t\xi.

Unlike the canonical vector field, which can be defined for any vector bundle, the canonical endomorphism

J:TTM\to TTM; \qquad J_\xi X := \operatorname{vl}_\xi(\pi_{TM})_*X, \qquad X\in T_\xi TM

is special to the tangent bundle. The canonical endomorphism J satisfies

\operatorname{Ran}(J)=\operatorname{Ker}(J)=VTM, \qquad \mathcal L_VJ= -J, \qquad J[X,Y]=J[JX,Y]+J[X,JY],

and it is also known as the tangent structure for the following reason. If (E,p,M) is any vector bundle 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 (TM,π_TM,M) of the base manifold, and J corresponds to the tangent structure of TM in this isomorphism.

There is also a stronger result of this kind ^[4] which states that if N is a 2n-dimensional manifold and if there exists a (1,1)-tensor field J on N that satisfies

\operatorname{Ran}(J)=\operatorname{Ker}(J), \qquad J[X,Y]=J[JX,Y]+J[X,JY],

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.

In any associated coordinate system on TM the canonical vector field and the canonical endomorphism have the coordinate representations

V = \xi^k\frac{\partial}{\partial \xi^k}, \qquad J = dx^k\otimes\frac{\partial}{\partial \xi^k}.

(Semi)spray structures

A 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, if in addition, [V,H]=H.

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.

Nonlinear covariant derivatives on smooth manifolds

The canonical flip makes it possible to define nonlinear covariant derivatives on smooth manifolds as follows. Let

T(TM\setminus 0) = H(TM\setminus 0) \oplus V(TM\setminus 0)

be an Ehresmann connection on the slit tangent bundle TM/0 and consider the mapping

D:(TM\setminus 0)\times \Gamma(TM) \to TM; \quad D_XY := (\kappa\circ j)(Y_*X),

where Y_*: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_X is a derivation in the module Γ (TM) of smooth vector fields on M in the sense that

$D_X(\alpha Y + \beta Z) = \alpha D_XY + \beta D_XZ, \qquad \alpha,\beta\in\mathbb R$ .
$D_X(fY) = X[f]Y + f D_XY, \qquad \qquad \qquad f\in C^\infty(M)$ .

Any mapping D_X with these properties is called a (nonlinear) covariant derivative ^[5] on M. The term nonlinear refers to the fact that this kind of covariant derivative D_X on is not necessarily linear with respect to the direction X∈TM/0 of the differentiation.

Looking at the local representations one can confirm that the Ehresmann connections on (TM/0,π_TM/0,M) and nonlinear covariant derivatives on M are in one-to-one correspondence. Furthermore, if D_X is linear in X, then the Ehresmann connection is linear in the secondary vector bundle structure, and D_X coincides with its linear covariant derivative.

References

↑ J.M.Lee, Introduction to Smooth Manifolds, Springer-Verlag, 2003.
↑ P.Michor. Topics in Differential Geometry, American Mathematical Society, 2008.
↑ 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
↑ D.S.Goel, Almost Tangent Structures, Kodai Math.Sem.Rep. 26 (1975), 187-193.
↑ I.Bucataru, R.Miron, Finsler-Lagrange Geometry, Editura Academiei Române, 2007.

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