During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below:

Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a submanifold of $M$ with induced metric $h_{ab}$ and extrinsic curvature $K_{ab}$.

We introduce a function $\tau$ on $\Sigma$ such that the two dimensional surfaces $\tau= \text{constant}$ in $\Sigma$ are
nested topological 2-spheres with the innermost surface reducing to a point. For each value
of $\tau$ let us assume that $S\subset \Sigma$ is one such surface and $\eta_a = \nabla_a\tau$ defines the normal to $S$ . The
unit normal is then given by $n_a = (\eta.\eta)^{-1/2}\eta_a$. Let $\xi^a:=un^a$ has the property that $\xi^a\nabla_a\tau = 1$.
Let $v^2:= \eta^a\eta_a$. So we have $\xi^a\nabla_a\tau = uv = 1$. So $u =1/v = (\eta^a\eta_a)^{-1/2}$. Next we consider the function $C(\tau)$ which for each value of $\tau$ is defined as
$$C(\tau):=\int_{S\subset \Sigma}(2\mathcal{R}-k^2)dA$$
where the integration extends over the surface $S$ and $\mathcal{R}$ and $k$ denote the scalar curvature and
the trace of the extrinsic curvature of the surface $S$ as a submanifold of $\Sigma$, respectively.
We note that the Gauss-Bonnet theorem implies that
$$\int_{S\subset \Sigma}\mathcal{R}dA=8\pi.$$
The trace of the extrinsic curvature $k$ of $S\subset \Sigma$ is defined as
$$k=\nabla_an^a.$$
**The rate of change of any quantity with respect to $\tau$ is its Lie derivative by $\xi^a$**. The rate of change of $k$ with respect to $\tau$ is then given by

$$\frac{\partial}{\partial \tau}k=\xi^b\nabla_b(\nabla_an^a)=\xi^b \nabla_a\nabla_bn^a-\xi^bR^a_{\;mab}n^m=\xi^b \nabla_a\nabla_bn^a-uR_{mb}n^bn^m.$$

Next after a calculation using the Gauss-Codazzi equation we arrive at the result that $$\frac{\partial}{\partial \tau}C(\tau)=\int_{S\subset \Sigma}(2kD^aD_au+ukk^{ab}k_{ab}-uk\mathcal{R}+ukR)dA.$$ where $D_a$ is the covariant derivative operator on the 2-surface $S$ with respect to the induced metric.

**Question1**: I don't understand why the bold statement is true. Can someone help me?

**Question2**: My efforts for deriving the last equation are failed. Can someone point me in the right direction? It is very important for me.

Thanks in advance for your time.