nLab tangent map

Given a differentiable map $f:M\to N$ of differentiable manifolds, for each $p \in M$ we have the differential $d f_p : T_p M \to T_{f(p)} N$ of $f$ at $p$. These maps define a unique tangent map $T f : T M \to T N$ of tangent bundles (as vector bundles) such that the diagram

$$\array{ T_p M &\stackrel{d_p f}\to & T_{f(p)}N\\ \downarrow &&\downarrow \\ T M &\stackrel{T f}\to & T N }$$

commutes for each $p$. The differential $d f_p$ of $f$ at $p$ (or sometimes the composition $T_p M\stackrel{d_p f}\rightarrow T_{f(p)}N\hookrightarrow T N$) is then denoted $T_p f$,

making $T$ into a functor (see differentiation).

category: geometry, differential geometry