I am quite new to the subject, please comment if something does not make sense.

Let $M$ be a Lie group. $L_x$ be left multiplication by $x\in M$.
I want to show that vector space $X:M \to TM$ defined as $X_x = T_1 L_x(v)$ is differentiable. ($v\in T_1M$)

If I am not mistaken, I need to argue the map \begin{equation} a \mapsto \left(a,\ T_{\varphi^{-1}(a)}\varphi (T_1L_{\varphi^{-1}(a)}(v))\right) = \left(a,\ T_1(\varphi\ \circ L_{\varphi^{-1}(a)})(v)\right) \end{equation} is differentiable ($a\in \mathbb{R}^n$, $\varphi$ is a chart). I am not sure how to argue tangent map is differentiable.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.