I've been trying to learn how to define the Levi-Civita connection as an Ehresmann connection, and I am getting quite confused. hoj201's answer on https://mathoverflow.net/questions/34088/a-geometric-interpretation-of-the-levi-civita-connection suggests that we can define the Levi-Civita connection as a certain Ehresmann connection of $\pi : TM\to M,$ but I am a little confused and am unsure their construction works.
- How does one define the Levi-Civita connection as an Ehresmann connection?
- What is the horizontal lift of a curve along $\pi$? If $\gamma : [0,1]\to M$ is a curve, is its horizontal lift just $\gamma' : [0,1]\to TM$?