How to choose basis vectors for a given metric to calculate curl of a vector in General Relativity? The space-time interval in General Relativity is $ds^2=g_{\mu\nu}dx^\mu dx^\nu$, where $g_{\mu\nu}$ is the metric tensor. If the space-time is stationary and axisymmetric, the corresponding geometry is described by the Kerr metric, where the non-zero metric coefficients are $g_{tt}$, $g_{rr}$, $g_{\theta\theta}$, $g_{\phi\phi}$ and $g_{t\phi}$. The off-diagonal term $g_{t\phi}\ne0$ implies the axisymmetry.
In the textbook by Landau & Lifshitz, the authors define the 3-dimensional metric tensor $$\gamma_{ij}=\left(-g_{ij}+\frac{g_{0i}g_{0j}}{g_{00}}\right),$$
such that the spatial distance is expressed as $dl^2=\gamma_{ij}dx^idx^j$. Then, the authors define the 3-dimensional vector $\mathbf{g}$ as $$g_i=-\frac{g_{0i}}{g_{00}}.$$
I am trying to calculate the curl of the vector $\mathbf{g}$, i.e., $\mathbf{\nabla}\times\mathbf{g}$, and for this I require to choose a suitable set of basis vectors. However, I am not able to understand how to find the basis vectors.
Could someone please suggest me any hints on how to proceed with the calculation?
 A: This might not be exactly, what you are looking for, but I am unsure, what you mean exactly by "finding basis vectors". If the task is to compute the curl, I suggest to proceed exactly as suggested by Ted Shifrin in the comments. Let me try to outline the concrete computation steps you need to do:
In the given situation, you are working on a 3-dimensional Riemannian manifold with metric $\gamma$. So the components of $\gamma$ can be expressed as $\gamma_{ij}$ with $i,j\in\{1,2,3\}$ in any given chart/coordinate system.  So you can choose any coordinate expression for the Kerr metric you like and compute $\gamma$ using the formula from Landau/Lifshitz.
In the same way you can use the other formula from Landau/Lifshitz you gave to compute the components of $\mathbf{g}$. Note that although Landau/Lifshitz call this a vector, it is really a 1-form or a co-vector field, because you compute $g_i$ with a lower index. The components of the actual vector field (contra-variant components) can be computed by $g^i = \gamma^{ij}g_j$, where you need the contra-variant metric components $\gamma^{ij}$. Compute them, e.g., by computing the matrix inverse of $(\gamma_{ij})$.
On the Riemannian manifold with metric $\gamma$, we can define the curl of a vector field $X$ with components $X^i$ as
$$ \nabla\times X := (\star_{\gamma}(\mathrm{d}X^\flat))^\sharp.$$
Here the musical isomorphisms $\flat$ and $\sharp$, just correspond to raising and lowering indices, i.e., $(X^\flat)_i = \gamma_{ij}X^j$ for a vector field $X$ and $(u^\sharp)^i = \gamma^{ij}u_j$ for a co-vector field or 1-form $u$. The operator $\mathrm{d}$ is the exterior derivative. For a 1-form $u$ the component expression for the resulting 2-form $\mathrm{d}u$ is just
$$(\mathrm{d}u)_{ij} = \partial_iu_j - \partial_ju_i.$$
$\star_\gamma$ is the Hodge star operator for the metric $\gamma$. In the 3-dimensional case here, it converts a 2-form into a 1-form. You can compute it for the metric $\gamma$, e.g., using the formula in this answer. For the 2-form $\mathrm{d}u$ in the 3-dimensional case here, it breaks down to
$$ (\star_\gamma(\mathrm{d}u))_k = \sqrt{\det(\gamma)}\gamma^{il}\gamma^{jm}\varepsilon_{klm}(\mathrm{d}u)_{ij}.$$
Putting everything together, what you might want to compute is:
$$ (\nabla\times\mathbf{g})^k =  \sqrt{\det(\gamma)}\gamma^{kr}\gamma^{il}\gamma^{jm}\varepsilon_{rlm}(\partial_ig_j - \partial_jg_i).$$
On a side note: I think what Landau/Lifshitz is doing there is called 3+1-splitting or lapse/shift-splitting in general relativity (cf. for example $\S$ 21.4 in Misner, Thorne, Wheeler: Gravitation). The Riemannian metric $\gamma$ will then only be independent of the time $t$ you split away if the spacetime is stationary, which of course is true for the Kerr metric.
