In my xAct project using
Mathematica, I encountered an issue while constructing a
×ばつ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.