Double tangent bundle

In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle T²M=T(TM) of the tangent bundle TM of a smooth manifold M. The double tangent bundle arises in the study of connections and second order ordinary differential equations, i.e. (semi)spray structures on smooth manifolds.

The double tangent bundle is closely related to the second order jet bundle, which is an object specifically designed to hold the "2nd order derivative information" of smooth functions on smooth manifolds.

Given a smooth map $f : M \to N$ there is an induced 1st-order derivative map $Tf : TM \to TN$ and so also a 2nd order derivative map $T^2 f : T^2 M \to T^2 N$.

The Lie Bracket of two vector fields on a manifold also has a formulation in terms of the double tangent bundle.

Canonical flip and associated coordinates

Unlike the tangent bundle, which only has one canonical vector space structure, the double tangent bundle has two vector bundle structures. Suppose $\pi\colon TM\to M$ and $\pi_2\colon T^2M\to TM$ are canonical projections for $TM$ and $T^2M$, respectively. Then $T^2M$ has vector bundle structures $(T^2M, \pi_2, TM)$ and $(T^2M, D\pi, TM)$. The canonical flip map $j$ is a diffeomorphism $j\colon T^2M\to T^2M$ that exchanges these vector space structures.

Let us give a coordinate representation for the canonical flip. Starting from local coordinates (U,φ) of the base manifold M we define the associated coordinates (TU,Tφ) on TM by

$\varphi:TU \to \varphi(U)\times \mathbb R^n \quad ; \quad (T\varphi)\Big(v^k\frac{\partial}{\partial x^k}\Big|_x\Big) := (x^1,\ldots,x^n,v^1,\ldots,v^n) \quad , \quad \varphi(x)=(x^1,\ldots,x^n).$

In terms of the coodinate basis of these associated coordinates the canonical flip reads as

$j\Big(X^k\frac{\partial}{\partial x^k}\Big|_Y + Z^k\frac{\partial}{\partial v^k}\Big|_Y\Big) = Y^k\frac{\partial}{\partial x^k}\Big|_X + Z^k\frac{\partial}{\partial v^k}\Big|_X \quad , \quad X = X^k\frac{\partial}{\partial x^k}\Big|_x\in T_xM \quad , \quad Y = Y^k\frac{\partial}{\partial x^k}\Big|_x\in T_xM.$

Let $(x^i, v^i, \xi^i, \eta^i)$ be the local coordinates on $T^2M$ associated with the local coordinates $(x^i,v^i)$ on TM associated with the local coordinates $(x^i)$ on M. Then the canonical flip reads as

$j(x^i, v^i, \xi^i, \eta^i) = (x^i, \xi^i, v^i, \eta^i).$

Canonical tensor fields on the tangent bundle

As for any vector bundle, the tangent spaces T_v(T_xM) of the fibres T_xM of (TM,π,M) can be identified with the fibres T_xM themselves. Formally this is achieved though the vertical lift

$vl:T_xM\times T_xM \to T(T_xM) \quad ; \quad vl(v,X)[f]:=(vl_vX)[f]:=\frac{d}{dt}\Big|_{t=0}f(x,v+tX)\quad , \quad f\in C^\infty(TM).$

The vertical lift is a natural vector bundle isomorphism vl:π^*TM→VTM from the pullback bundle of (TM,π,M) over π:TM→M onto the vertical tangent bundle VTM:=Ker(π_*)⊂TTM.

In terms of the vertical lift we can define two canonical tensor fields on (TTM,π₂,TM), the canonical vector field

$V:TM\to TTM \qquad ; \qquad V_v := vl_vv$

and the tangent structure or canonical endomorphism

$J:TTM\to TTM \qquad ; \qquad J_vX := vl_v\pi_*X \qquad , \qquad X\in T_vTM.$

In the local coordinates (TU,Tφ) on TM associated to (U,φ)these canonical tensor fields have coordinate representations

$V = v^k\frac{\partial}{\partial v^k} \qquad ; \qquad J = dx^k\otimes\frac{\partial}{\partial v^k}.$

The canonical endomorphism J satisfies

$Ran(J)=Ker(J)=VTM \quad , \quad \mathcal L_VJ= -J \quad , \quad J[X,Y]=J[JX,Y]+J[X,JY],$

and in a certain sense the existence of such tensor field J on a 2n-dimensional manifold implies that the manifold is a (part of) a tangent bundle of some n-dimensional manifold.

The canonical vector field can also be defined as follows. For v∈TM define f_v:R→TM by f_v(t)=tv, where tv∈TM is the scalar multiplication. Then V_v:=Tf_v(1,1)∈TTM, where we identify TR with R² in the standard way.

(Semi)spray structures

A Semispray structure on a smooth manifold M is dy definition a smooth vector field H on TM \0, or in other words, a smooth section of the deleted double tangent bundle (T(TM \0),π₂,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 ivariant 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.

Connections on the tangent bundle

Connections on $M$ has a formulation in terms of a projection map $T^2 M \to TM$ due to Ehresmann. Let $\pi_2 : T^2 M \to TM$ and $\pi_1 : TM \to M$ be the bundle projections for the double tangent bundle and tangent bundle respectively.

See also

• Spray (mathematics)

References

• D.S.Goel. Almost tangent structures, Kodai.Math.Sem.Rep. 26 (1975) 187-193.
• P.Michor. Topics in Differential Geometry, American Mathematical Society (2008).
• Guillemin and Pollack. Differential Topology, Prentice-Hall (1974).