Hello,

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!

Ben

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!