# Parallel transport, vector fields

Let $\mathbf{X}$ vector field along curve $\gamma: [a,b] \longrightarrow \mathcal{M}$. We can parallel transporated each vector $\mathbf{X}(t)$ in the point $\gamma(a)$. That is, we can get a set of vectors $P_t^a \mathbf{X}(t)$ in the tangent space $T_{\gamma (a)} M$. I read in the book that this set can be viewed as a vector field along map $P \circ \gamma : [a, b] \longrightarrow T_{\gamma (a)} M$.

I did not understand it. What exactly is a vector field? The map $P_t^a$? And what is the map P? And what is codomain? Codomain of vector field should be a tangent space of codomain map...

Sorry for my English

Here is this book, but there written on russian

A vector field is a smooth section $s:M\to TM$ of the tangent bundle. So if $\pi:TM\to M$ is the projection of the tangent bundle of $M$ onto $M$(takes each tangent space $T_pM$ to $p$), we have $\pi\circ s=id_M$.

Here is a reference on parallel transport.

Secondly, the codomain may be familiar to you as the range of or target space of a function. ( Though be careful: range of $f$ can mean image of $f$.)

• Thanks for the answer. I know what a vector field and parallel translation are. I know what is $P_a^b$. I'm a little confused about $P$ and $P \circ \gamma$. And why $P_t^a$ can be viewed like vector field along $P \circ \gamma$. Commented Aug 29, 2018 at 15:29
• How to copmose $P$ with $\gamma$? $P$ is defined on $\gamma ([a, b])$? Commented Aug 29, 2018 at 16:53
• Sure. $\gamma$ is a curve on $M$; and $P$ goes from $M$ to $TM$.
– user403337
Commented Aug 29, 2018 at 16:57
• $P: M \to TM, \gamma(t) \mapsto v \in T_{\gamma(t)}$? Commented Aug 29, 2018 at 17:09
• If it is right, why $P \circ \gamma$ act in $T_{\gamma(a)}$? Commented Aug 29, 2018 at 17:22