Tangent functor - Imported Knowlpedia Content

Definition

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. The functorial formulation and its natural transformations are treated in Kolář–Michor–Slovák, Chapter VI.

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.