Tangent functor - Knowlpedia

The tangent functor is the covariant functor

T:\mathbf{Man}\longrightarrow\mathbf{Man}

on the category of smooth manifolds. It sends a manifold $M$ to its tangent bundle $TM$ , regarded as a smooth manifold, and a smooth map $f:M\to N$ to the smooth map

Tf:TM\longrightarrow TN,\qquad v_p\longmapsto df_p(v_p),

induced by the differential of $f$ . The chain rule gives $T(g\circ f)=Tg\circ Tf$ , while $T(\operatorname{id}_M)=\operatorname{id}_{TM}$ ; these are precisely the identity and composition axioms required of a functor.

Bundle structure and natural maps

For each $f:M\to N$ , the map $Tf$ is a bundle map over $f$ :

\pi_N\circ Tf=f\circ\pi_M.

Consequently, the bundle projections $\pi_M:TM\to M$ assemble into a natural transformation $T\Rightarrow\operatorname{Id}_{\mathbf{Man}}$ . The zero sections $0_M:M\to TM$ similarly assemble into a natural transformation $\operatorname{Id}_{\mathbf{Man}}\Rightarrow T$ .

Products and isomorphisms

There is a canonical diffeomorphism $T(M\times N)\cong TM\times TN$ , under which $T(f\times g)=Tf\times Tg$ . If $f$ is a diffeomorphism, then $Tf$ is a diffeomorphism with inverse $T(f^{-1})$ . Thus tangent spaces and differentials form one coherent construction, rather than unrelated pointwise assignments.

Conventions and scope

This knowl concerns the ordinary first tangent functor on finite-dimensional smooth manifolds. Iterated tangent bundles $T^kM$ , higher-order tangent functors, tangent functors on manifolds with corners, and tangent constructions in synthetic or infinite-dimensional differential geometry require additional conventions.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 3, tangent vectors, tangent bundles, and differentials.
Ivan Kolář, Peter W. Michor, and Jan Slovák, Natural Operations in Differential Geometry, Springer, 1993. DOI record. Relevant: Chapter VI, tangent functors and natural transformations.