tangent map

Given a differentiable map f:MNf:M\to N of differentiable manifolds the differentials? df p:T pMT f(p)Md f_p: T_p M\to T_{f(p)}M of ff at varying points pMp\in M define a unique map Tf:TMTNT f : T M\to T N of tangent bundles as vector bundles such that each pMp\in M the diagram

T pM d pf T f(p)N TM Tf TN\array{ T_p M &\stackrel{d_p f}\to & T_{f(p)}N\\ \downarrow &&\downarrow \\ T M &\stackrel{T f}\to & T N }

commutes. The differential df pd f_p at pp (or sometimes the composition T pMd pfT f(p)NTNT_p M\stackrel{d_p f}\rightarrow T_{f(p)}N\hookrightarrow T N) is then denoted T pfT_p f, making TT into a functor, see differentiation.

Created on August 7, 2016 at 08:54:53. See the history of this page for a list of all contributions to it.