I'am currently trying to understand the paper:
https://epub.uni-regensburg.de/23578/1/MP171.pdf
The point where I'am stucked at is in the proof of Theorem 4.6.
You dont need to read the whole article, here are the informations one need:
We consider a Lorentzian Manifold M.
Let $ \tau:M\to\mathbb{R} $ be a timelike distance function with $ \textbf{n}:=-\text{grad}\:\tau $. We define the shape operator by $ S(X)=\nabla_{X}\textbf{n} $ for $ X\in \Gamma^{\infty}(TM) $. Also consider the integralcurve $ \gamma:I\to M $ of $ \textbf{n} $.
In the paper it was shown that the shape operator is actually solving:
$ \nabla_{\textbf{n}}S+S^{2}+R_{\textbf{n}}=0 $ (*)
where $ R_{\textbf{n}}(X):=R(X,\textbf{n})\textbf{n} $ denotes the Riemann curvature tensor and $ \nabla$ is the Levi-Cevita Connection on $TM$.
My Question is now the following:
How can I use the pullback connection $ \nabla^{\gamma^{*}TM} $ on $ \gamma^{*}TM $ to rewrite the equation (*) in the following form:
$ \nabla^{*}_{\partial_{t}}\gamma^{*}S+(\gamma^{*}S)^{2}+\gamma^{*}R_{\textbf{n}}=0 $ (**)
I know the notation in equation (**) is really unlucky, but at the moment i don't how to deonote it better.