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I'm looking for a resource regarding the chain rule for covariant derivatives.

The closest version I've seen is the one regarding differentials of maps: If F and G are differential maps, then $d(G \circ F)_p=dG_{F(p)} \circ dF_{p}$

(From deCarmo's Differential Geometry)

I'm not positive this extends to covariant derivatives, although I feel it should, as the differential applied to a vector $v$ and the covariant derivative in the direction $v$ differ only by a projection onto the tangent plane. Could someone please point me towards the right resource? Thanks so much!


Edit: Regarding the comments below by Ryan and Deane:

I'm thinking about a covariant derivative (using the standard Levi-Civita connection) on a surface $S$ embedded in $\mathbb{R}_3$. Let $dN$ be the shape operator (differential of the Gauss map) and let $u \in T_p(S)$. Say $\alpha(t)$ is a curve on the surface with $\alpha'(0) = u$. Let $v(t)$ be a vector field defined on $\alpha(t)$. Then, the equation I am interested in would be: $\nabla_u (dN(v(t))) = (\nabla_u dN) (\nabla_u v(t))$. That is, the R.H.S is the covariant derivative (in the tensor sense) of $dN$ applied to the tangent vector $\nabla_u v$. I would like to know whether equation is true, or if there is a similar type of rule. Thanks again for your time!

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closed as not a real question by Ryan Budney, Willie Wong, Deane Yang, Will Jagy, Mariano Suárez-Alvarez Jun 18 '11 at 5:48

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Covariant derivatives usually apply to objects like sections of fibre bundles when you have some kind of connection. So what kind of compositional operation do you want to consider? – Ryan Budney Jun 17 '11 at 21:30
Agree with Ryan. Please cite a specific formula you think might be true. – Deane Yang Jun 17 '11 at 22:25
I'm not sure why this was closed as "not a real question". There actually is a very useful chain rule for the covariant derivative and it has to do with pullback bundles. If $\pi : E \to N$ is a vector bundle (e.g. $TN$), $\nabla^E$ is a covariant derivative on $E$, $\sigma$ is a section of $\pi$, and $\phi : M \to N$ is a map, then $\nabla^{\phi^* E} (\sigma \circ \phi) = (\nabla^E \sigma \circ \phi) \cdot T\phi$. This can be seen trivially using the "connection map" formulation $\nabla^E \sigma := \kappa \cdot T\sigma$. If someone re-opens this question, I'd gladly post a full answer. – Victor Dods Dec 16 '11 at 1:18