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I am interested in the Christoffel symbol. So I found Christoffel's 1869 paper and The Road to Reality by Roger Penrose. These days, the Christoffel symbol has been established as a covariant derivative, but Christoffel did not know the concept of covariant derivative and parallel transport.

So, I wondered what concept Christoffel used to create the Christoffel symbol at the time.

When I read the book The Road to Reality, I think that Lie algebra was use. What exactly is the concept behind the Christoffel symbol?

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    $\begingroup$ Don't they show up naturally when you compute derivatives in an arbitrary coordinate system with a non-constant metric? $\endgroup$
    – Frank
    May 26 at 14:02
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    $\begingroup$ Consider reading chapters IV and V of Volume 2 of Spivak's "Comprehensive Introduction to Differential Geometry": Among other things, he discusses the origin of Christoffel symbols. $\endgroup$
    – Moishe Kohan
    May 26 at 14:43
  • $\begingroup$ Pretty sure it would be from gauss. By the way, I finished a large portion of that book already, so if you want to discuss it, feel free to send me a message on the stackexchange chat. $\endgroup$
    – tryst with freedom
    May 26 at 22:51

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