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I'm working my way through Ivey and Landsberg, Cartan for Beginners, and I'm working on 2.6.13.2(b). After defining, for $X \in \Gamma(TM)$ a vector field, with section $s:M \rightarrow \mathcal{F}_{ON}(M)$ a (local) orthonormal coframing, we have $X=X^ie_i$ for some functions $X^i$. Define the covariant derivative of $X$ to be $$\nabla X = (dX^i + X^j\eta^i_j)\otimes e_i \in \Omega^1(M,TM) = \Gamma(TM\otimes T*M),$$ with $\eta^i$ and $\eta^i_j$ being pulled back via $s$.

[...]

(b) For $Y \in \Gamma(TM),$ we define $\nabla_YX := Y\neg\nabla X = (dX^i + X^j\eta^i_j)(Y)e_i.$ Show that $\nabla_Y(fX) = f\nabla_YX + X(f)Y$ for $f\in C^{\infty}(M)$....

Here is my question: I've seen multiple explanations (self-learner) of the connection forms defined here as tautological 1-forms on $T\mathcal{F}_{ON}$. I've seen multiple definitions of the connection, Cartan, Koszul, Ehresmann, and one of them which defined it in terms of a splitting of an exact sequence involving the vertical bundle over $\mathcal{F}$ also said that for adapted coordinates, $\eta^i_j(Y)$ becomes $\Gamma^i_{kj}Y^k$.

What is the relationship between the different definitions, and does that latter work for all three different definitions of the connection (one definition, Koszul in Darling, had created $\Gamma(M)$ as a local vector valued form that pulls back from $\mathbb{R}^n$ as well, so it seems like it can be defined generally and consistently enough but I'm very unsure).

I do remember with, say, integrals, the increasingly abstract definitions (Riemann, Stiltjes, Lesbesgue) were sort of nested so that the less general definition didn't change value when one moved to increasing generality, but I've seen no such explanation for all these characterizations of connections.

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    $\begingroup$ I'm not going to write out a whole lecture here, but you might look, for example, at Volume 2 of Spivak's 5-volume opus, where he talks about the four main definitions (classic indices, Koszul, moving frames, and principal bundles) and relates them all excruciatingly carefully. $\endgroup$ – Ted Shifrin Jul 19 '16 at 16:36
  • $\begingroup$ Thanks. Would that be Volume 2, Ch.8 and the addendums? I will read it more carefully than I must have. $\endgroup$ – hkr Jul 19 '16 at 17:11
  • $\begingroup$ Well, as he goes along from Chapter 5 through Chapter 8, he relates each notion to the previous ones. I'll be happy to answer more specific questions, but I can't write out a week's worth of lectures here to answer all of your general question. :) (Actually, I have answered a few such questions in the past here — much more narrowly constructed questions. You can certainly look at some of my differential geometry answers.) $\endgroup$ – Ted Shifrin Jul 19 '16 at 17:20
  • $\begingroup$ Understood. My most specific question is how generally one can relate the forms $\eta^i_j$ derived from the tautological 1-form and the bundle valued approach to the Christoffel symbols. It seems like the Christoffel symbols are more limited, but they can also be given a bundle valued form definition, so I'm not sure what they are more limited to. $\endgroup$ – hkr Jul 19 '16 at 17:30
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    $\begingroup$ The classical Christoffel symbols come from working in local coordinates, rather than with an orthonormal moving frame. However, we can (and sometimes do) write $\eta^i_j = \sum \Gamma^i_{jk}\omega^k$ (where the $\omega^k$ are the dual coframe to $e_k$). $\endgroup$ – Ted Shifrin Jul 19 '16 at 18:00

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