Define $g$ to be the metric on $R^3$ with components $$g_{11}=g_{22}=g_{33}=(x^1)^2+1, g_{ij}=0 \mbox{ for } i\neq j.$$ What are the geometric implications of imposing this metric on $R^3?$ Is this the same with the Euclidean space?

More generally, let $f:R^n \to (0, \infty)$. Then $g$ with components $$g_{ij}=\delta_{ij}f$$ is a metric on $R^n$, right? How is this different from the Euclidean space?

Thanks much, in advance.