Bdtyk

3 1 $begingroup$ I must have read and re-read introductory differential geometry texts ten times over the past few years, but the "torsion free" condition remains completely unintuitive to me. The aim of this question is to try to finally put this uncomfortable condition to rest. Ehresmann Connections Ehresmann connections are a very intuitive way to define a connection on any fiber bundle. Namely, an Ehressmann connection on a fiber bundle $Erightarrow M$ is just a choice of a complementary subbundle to $ker(TE rightarrow TM)$ inside of $TE$ . This choice is also called a horizontal bundle. If we are dealing with a linear connection, then $E=TM$ , and the Ehresmann connection is a subbundle of $TTM$ . This makes intuitive sense -- basically it's saying that for each point in $TM$ it tells