Iterated tangent bundle - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold. Its iterated tangent bundles are defined recursively by

T^0M=M,\qquad T^{r+1}M=T(T^rM)\quad(r\geq0),

using the tangent functor. In particular, $T^2M=T(TM)$ , usually written $TTM$ , is the second tangent bundle. If $M$ has dimension $n$ , then $T^rM$ has dimension $2^rn$ . For a smooth map $f:M\to N$ , repeated differentiation gives $T^rf:T^rM\to T^rN$ ; hence each fixed iterate $T^r$ is again a covariant functor on smooth manifolds.

The double vector-bundle structure of

TTM

The second tangent bundle has two natural projections to $TM$ :

\tau_{TM}:TTM\to TM,\qquad T\tau_M:TTM\to TM,

where $\tau_M:TM\to M$ is the tangent-bundle projection. Each projection makes $TTM$ a vector bundle over $TM$ , and the two structures satisfy compatibility axioms. Thus $TTM$ is the basic example of a double vector bundle, not merely an ordinary vector bundle with duplicated notation; see Mackenzie, Chapter 9.

Canonical involution

A smooth two-parameter map $\gamma(s,t)$ into $M$ determines an element of $TTM$ by differentiating first in one parameter and then in the other. Interchanging the parameters defines the canonical involution

\kappa_M:TTM\longrightarrow TTM,\qquad \kappa_M^2=\operatorname{id}_{TTM}.

It exchanges the two vector-bundle projections. In induced coordinates $(x,v;\dot x,\dot v)$ , it has the form

\kappa_M(x,v;\dot x,\dot v)=(x,\dot x;v,\dot v).

This construction is natural in $M$ .

Examples and conventions

For a vector space $V$ , translation gives $TV\cong V\times V$ and $T^rV\cong V^{2^r}$ , although the canonical projections still distinguish the factors. For a general manifold no global product decomposition exists.

References

Ivan Kolář, Peter W. Michor, and Jan Slovák, Natural Operations in Differential Geometry, Springer, 1993. Springer DOI record. Relevant: Chapter VI, iterated tangent functors and natural transformations.
Kirill C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005. Cambridge DOI record. Relevant: Chapter 9, double vector bundles and the canonical involution of $TTM$ .