Pullback bundle, connections

Question

First of all, I know that my question is quite long. I still would really appreciate any help because I am kind of stuck with this topic, I really try to express my question as clear as possible! Also, sorry for the language!

Let $(V,g), (M,h)$ be semi-Riemannian manifolds, $u: V \rightarrow M$ continous. Let $dim \ V=n+1, dim \ M=d$. Let $U \subset V$ be an open set with coordinates $x_{\alpha}$, s.t. $u(U)\subset M$ with coordinates $y^A$.

Next, let $E= T^{*}V \otimes TM$ be the vector bundle such that $E_x=T_xV \otimes T_{u(x) M}$.

In a paper is says that "The covariant derivative ∇ acting on sections of E has for coefficients the connexion coefficients of the Riemannian connexion of g and the pull back by u of the Riemannian connexion of h" and there is a local expression for the covariant derivative, where $f \in \Gamma(E)$:

$\nabla_{\alpha} f^A_{\beta}(x)= \partial_{\alpha} f^A_{\beta}(x)- \Gamma^{\mu}_{\alpha \beta}(x) f^{A}_{\mu}(x)+ \partial_{\alpha}u^B(x) \Gamma^A_{BC}(u(x))f^C_{\beta}(x)$

where $\Gamma^{\mu}_{\alpha \beta}$ are the components of the covariant derivative of $g$ and $\Gamma^A_{BC}(u(x))$ are the components of the covariant derivative of $h$.

Now I am trying to understand how this works. My thoughts so far:

Let $f \in E$. Then $f$ is of the form

$f_x=\underbrace{(\sum\limits_{\alpha=1}^{n+1} f_{\alpha}(x) \partial_{\alpha}|_x)}_{s} \otimes \underbrace{(\sum\limits_{A=1}^{d} \tilde{f}_A(x) \partial_A|_{u(x)})}_{t}$

where $f_{\alpha}, \tilde{f}_{A}:U \rightarrow \mathbb{R}$.

Is that correct so far?

Now it also holds that $\nabla (s \otimes t)= (\nabla s) \otimes t + s \otimes (\nabla t)$

So now I only need to understand how $\nabla s$ and $\nabla t$ looks like.

1. $\nabla_{\alpha} s= \sum\limits_{\beta=1}^{n+1} (\partial_{\alpha}(f_{\beta}) \partial_{\beta} + f_{\beta} \sum\limits_{\mu=1}^{n+1} \Gamma^{\mu}_{\alpha \beta} \partial_{\gamma})$ Is that correct?

2. Now for $\nabla t$, I don't really know what to do with it...I know the definition of a pullback bundle but I don't know how to express its covariant derivative in local coordinates. Can someone help me with this step?

Thanks in advance!

Answer 1

Since $f$ is a section of $T^{*} V \otimes u^{*}(TM)$, it can be written locally as

$$ f = f_{\beta}^{A} \, dx^{\beta} \otimes u^{*}(\partial_{A}) $$

where the $x^{\beta}$ are coordinates on $V$ and the $x^{A}$ are coordinates on $M$ and we are using Einstein summation so that the sums won't clutter our expression (note that this is not what you wrote as you need to use the $dx^{\beta}$ and not $\partial_{\beta}$).

Then the covariant derivative in the direction $\partial_{\alpha}$ is given by

$$ \nabla_{\partial_\alpha} f = \nabla_{\partial_\alpha} \left( f_{\beta}^{A} \, dx^{\beta} \otimes u^{*}(\partial_{A}) \right) = \\ \left( \partial_{\alpha} f^A_{\beta} \right) dx^{\beta} \otimes u^{*}(\partial_{A}) + f^A_{\beta} \left( \nabla_{\partial_\alpha} dx^{\beta} \right) \otimes u^{*}(\partial_A) + f^A_{\beta} dx^{\beta} \otimes \left( \nabla_{\partial_{\alpha}} u^{*}(\partial_A) \right) $$

where we used the product rule to move from the second expression to the third. Now use the definition of the connection on $T^* V$ and on $u^{*}(TM)$ to write the second and third expression in terms of the Christoffel symbols and gather coefficients to get formula you want.