Problem in Writing Gaussian Null Coordinate in xAct

[Ashes Modak]

In my xAct project using Mathematica, I encountered an issue while constructing a 4×4 matrix using CTensor on a 4-dimensional manifold "M1" with chart "Old", whose coordinates are (v,r,x_{1},x_{2}).x_1, and x_2 spans a 2 dimentional spatial space whose metric is h_{ij}.The components of this spatial metric specifically h_{11}​,h_{12}​,h_{21}​,h_{22​}. So i want to define h_{ij} to be metric in this smaller spatial manifold, so that it can contract and lower and raise indices of tensors defined in the spatial manifold. But I can't define h_{ij} as a metric.

The whole metric is given by,

ds^{2}=2dvdr-r^{2}X(r,v,x^{i})dv^{2}+2rw_{i}(r,v,x^{i})dvdx^{i}+h_{ij}(r,v,x^{i})dx^{i}dx^{j}

I want to define it in xAct and find the Christoffel, Ricci, and Riemann tensors.