Let's consider a plane which coordinates are $x^1,x^2$. Then, consider the basis given by ${\partial \vec P/\partial x^i}$. Accordingly to this choice of coordinates and basis there will be a metric tensor in every point $T(x^i)$.
Is it possible to find coordinates such that the basis $\partial \vec P/\partial x^i$ change from point to point but the metric tensor is constant,$T(x^i)=constant$?
For example, with cartesian coordinates the metric tensor is constant but the basis is also constant everywhere so it does't satisfy the requirement.