Applications of Koszul's formula other than the fundamental lemma of Riemannian geometry

I'm wondering what else one can do with Koszul's formula

$$2\langle\nabla_XA,B\rangle = X\langle A,B\rangle-B\langle X,A\rangle + A\langle X,B\rangle - \langle A,[X,B]\rangle + \langle[B,X],A\rangle - \langle B,[A,X]\rangle$$

beyond proving existence and uniqueness of the Levi-Civita connection. I haven't yet seen anybody using it for anything else, which would be quite curious.

Here's a pretty and simple example. I don't know if it is known...

Let $\nabla^{LC}$ be the Levi-Civita connection and $\nabla$ be some other metric connection with torsion $T$. Then

$$2\langle\nabla_XA,B\rangle - 2\langle\nabla_X^{LC}A,B\rangle= \langle T(A,X),B\rangle -\langle T(A,B),X\rangle - \langle T(B,X),A\rangle$$

An application of this would be to compute the LC-connection for the metric $\langle K\cdot,K\cdot\rangle$ in terms of the endomorphism $K$ and the LC-connection for $\langle \cdot,\cdot\rangle$. The computation starts with the connection $K^{-1}\nabla^{LC} K$, which is metric for $\langle K\cdot,K\cdot\rangle$...

I can't guarantee correct letters and signs :-)

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Riemannian geometry IS the study of the Levi-Civita connection and the Koszul formula is an explicit expression for the connection. So ultimately anything you do in Riemannian geometry is bound to be using it at some level. If, on the other hand, you mean what other formulae you can get by formally manipulating the Koszul formula, then there are plenty, but I would not necessarily call them "applications". Killing's equation comes to mind, as well as the O'Neill formulae for submersions, the basic formulae for homogeneous geometry,... Pick any book on differential geometry: e.g., do Carmo. –  José Figueroa-O'Farrill Mar 7 '11 at 11:18
Indeed. Except when the glory breaks down and coordinates get introduced (or used implicitly). (If not, wait till play with Laplacians begins and frames introduced...) BTW, one application of my application would be the coordinate representation of the LC connection. PLEASE NOTE: I'm far away from the library and made it there only a few times this century - and it was open a few of these occasions. But I've seen and possibly remember quite a few of these books, e.g. none on Ricci flow without ungeometricly resorting to coordinate stuff when things get most basic –  Martin Gisser Mar 7 '11 at 15:09
So let me get this straight: you make a distinction between the Koszul formula and the formula for the Christoffel symbols? I don't think there is any conceptual difference between the two. I suppose it's a question of how much pain are you willing to endure in order to keep everything coordinate-free. Analysts as a general rule seem to have a low pain threshold :) –  José Figueroa-O'Farrill Mar 7 '11 at 17:40
Hm, yes I'm actually trying to avoid pain. I might perhaps have an infinitude of indices (symbolic or not) or try to keep track of the tensor product rule (after all, it's tensor calculus) and not get lost (or waste sparse IQ) in the debauch of indices. Why else Koszul connections? Of course Christoffel stuff and Koszul covariant derivatives (plus Cartan's calculus, Penrose spaghetti, etc. pp.) are all about the same thing, but at least there should be some motivation to stay in one concept (plus, avoid coordinates (it could be a ringed space) (or the coordinates being unknown or irrelevant)). –  Martin Gisser Mar 7 '11 at 18:05
Quite a borderline comment but I find it funny. There is (at least) one thing I don't know how to prove without coordinates. Whena metric evolves by the Ricci flow, the evolution of the curvature operator is given by : $\partial_t R=\Delta R + R^2+R^#$, where $R^#$ can be defined in a coordinate free way using the Lie algebra structure of $\Lambda^2 TM$. A substantial part of the proof can be done without coordinates (as in Toppong lectures) but I never saw anyone that goes from the expression ypu found in Topping book to the Lie algebraic expression without coordinates. –  Thomas Richard Mar 10 '11 at 7:51

I've meanwhile found out that my innocent example ...

1.) ... amounts to a rediscovery of Schouten's contorsion tensor. (See e.g. this note).

This concept is important in torsion gravity (which isn't as exotic as it may sound - except for some charlatanry derived from it...) My example equation seems to express the equivalence principle (cf. link eq. (255)). See also Rodrigues & Oliveira: The many faces of Maxwell, Dirac and Einstein equations.

As I've already hinted here, contorsion stuff (or what I would term the contorsion operator) can also occur in computations with torsion-free connections.

2.) ... leads to a generalization of the fundamental "lemma", a.k.a. Levi-Civita theorem: For any given vector-valued 2-form $T$ there exists a unique metric connection with torsion $T$.

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