So I am reading this paper. The metric is Eq. (3.1) reads:

$$ds^2=r^{2}d\tau^{2}+n^{2}dr^{2}+\Bigg[\gamma_{ij}(v)+2r^{n}c^{1-n}\bigg(\cos\tau

k^{(1)}_{ij}(v)+\sin\tau

k^{(2)}_{ij}(v)\bigg)\Bigg]dv^{i}dv^{j}$$

The definition of extrinsic curvature can be written as:

$$K^{a}_{b}=\nabla_b n^{a}=\partial_{b}n^{a}+\Gamma^{a}_{\ bc}n^{c}$$

where Latin indices are running from 1 to d-1. My understanding tells me this is something standard, by which I mean extrinsic curvature is being defined on the hypersurface, hence it is d-1. Yet below Eq. (3.3), the paper calls for $K^{\mu}_{\nu}$ where greek indices are running 1 to d. However, later on above Eq. (4.6), the paper mentions $K^{a}_{b}$. Is this a typo? Also how does one calculate the normal vector n's? Specifically can anyone guide how to obtain $K^{\tau}_{\tau}=1/n\epsilon$ as mentioned below Eq. (3.3).