On the completions of the spaces of metrics on an open manifold
For a smooth manifold \(X\) the function \(d_0\) defined by: \[ d_0(g,g')=\sup_{x\in X}\sup_{Y\in T_x X-\{0\}} {|g_x(Y,Y)-g_x'(Y,Y)|\over g_x(Y,Y)+g_x'(Y,Y)} \] is a complete metric on the space of complete \(C^0\) metrics on \(X\). This metric is then used to show that a space of strictly positive definite smooth metrics with a natural uniform structure is complete.
scientific article; zbMATH DE number 920817