tangent map

Given a differentiable map $f:M\to N$ of differentiable manifolds the differentials? $d f_p: T_p M\to T_{f(p)}M$ of $f$ at varying points $p\in M$ define a unique map $T f : T M\to T N$ of tangent bundles as vector bundles such that each $p\in M$ 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. The differential $d f_p$ 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: geometrydifferential geometry

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