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In the skript of my professor we define the horizontal lift in a fiber bundle as follows: \begin{align} ^{hor}: TM\times_ME\rightarrow TE \end{align} with \begin{align} (Tpr\times \pi_{TE})\circ^{hor}=id_{TM\times_M E} \end{align} where \begin{align} \pi_{TE}: TE \rightarrow E \end{align} is the tangent bundle projection. With $TM\times_M E$ we usually denote the fiber product. I have two questions:

  1. If I understood correctly, the horizontal lift is a map that "lifts" tangent vectors of the base manifold to (horizontal) tangent vectors of the total space. So $^{hor}$ should be a map $^{hor}: TM \rightarrow TE$. Why did he define it as a map $^{hor}: TM\times_ME\rightarrow TE$?
  2. Im not used to the notation $Tpr\times \pi_{TE}$ and I can't find any help in the skript, too. What could he mean with $\times$ between two maps here?
Aralian
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    Regarding point (1), you need to know “which height to lift to”, that’s why you also need the $E$ factor. For point (2), simply note that applying $T(\text{pr})$ gives you something in $TM$, and applying $\pi_{TE}$ gives you something in $E$ (and both will be over the same base point in $M$). You now stick these two together in a tuple to get an element of $TM\times_ME$. Anyway, this is too condensed. See the first part of Intuition behind connection 1-forms and Ehresmann connections for more details (I call the map $L$ there, for ‘lifting’). – peek-a-boo Sep 18 '23 at 09:13

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