I'm reading this paper and I'm stuck in the lemma $2.1$. There are three points that I can't understand.
My first doubt is why the following equality is valid?
$$\frac{\partial^2 p^{\alpha}}{\partial s^2} - \Gamma^0_{00} \frac{\partial p^{\alpha}}{\partial s} + \tilde{\Gamma}^{\alpha}_{\beta \gamma} \frac{\partial p^{\beta}}{\partial s} \frac{\partial p^{\gamma}}{\partial s} = - h^i_{00} \nu^{\alpha}_i$$
I know that Gauss-Weingarten relations in this context is
$$\frac{\partial^2 p^{\alpha}}{\partial s^2} = \left( \frac{\partial^2 p^{\alpha}}{\partial s^2} \right)^T + \left( \frac{\partial^2 p^{\alpha}}{\partial s^2} \right)^{\perp},$$
but I can't see how obtain the tangent component and why the normal component has a minus sign.
My second doubt is why the following equation is valid?
$$H_ i = e^{-2f} \left( \frac{\partial^2 p^{\alpha}}{\partial s^2} \nu^{\beta}_i \tilde{g}_{\alpha \beta} + \tilde{\Gamma}^{\alpha}_{\beta \gamma} \frac{\partial p^{\beta}}{\partial s} \frac{\partial p^{\gamma}}{\partial s} \nu^{\gamma}_i \tilde{g}_{\alpha \gamma}\right)$$
I think the $H_i$ is the mean curvature with respect to $\nu_i$, but I can't see why $- \Gamma^0_{00} \frac{\partial p^{\alpha}}{\partial s}$ doesn't appear here, but $\tilde{\Gamma}^{\alpha}_{\beta \gamma} \frac{\partial p^{\beta}}{\partial s} \frac{\partial p^{\gamma}}{\partial s}$ appear.
Finally, I can't see how $H_i$ becomes $\tilde{\Gamma}^i_{00}$ and why this is equal to $\nabla_i f$?
Thanks in advance!