# Pullback bundle, connections

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?

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.

• BTW, I suspect there is a sign error in your formula. The last term should be plus and not minus. – levap Jun 13 '20 at 19:05
• Thank you very much for your answer, it already cleared up a lot of questions. I also fixed the sign error. I now tried to continue: It holds, that $\nabla_{\partial_{\alpha}} d x ^{\beta} = \sum\limits_{\mu=1}^{n+1} \Gamma^{\mu}_{\alpha \beta} d x^{\mu}$. Now I don't see where the $f^A_{\mu}$ in the expression from the paper comes from here. (Sorry for not using the Einstein convention, it still kind of confuses me) – User1 Jun 16 '20 at 10:50
• My result would be $\nabla_{\alpha} f^A_{\beta}= \partial_{\alpha} f^A_{\beta}(dx^{\beta} \otimes u^{*}(\partial_A)) + f^A_{\beta} (\Gamma^{\mu}_{\alpha \beta} d x ^{\mu}) \otimes u^{*}(\partial_A)+ f^A_{\beta} (dx^{\beta}) \otimes (\partial_{\alpha} u^{\beta} \Gamma^C_{BA} (u(x)) u^{*}(\partial_C))$. Is there already a mistake here or do I just miss why it is the same as in the paper? – User1 Jun 16 '20 at 11:22