While studying some differential geometry, a thought crossed my mind that I am sure has been considered before, but I cannot find a reference for it.
What can be said about spaces for which the metric has pathological behaviour? I know that this is playing fast and loose with the terminology, but bear with me.
In particular I am curious about generalised metrics that change signature. My rough idea of an example is $\mathbb{R}^2$ where we have a notion of length that is asymptotically Minkowskian, but near to the origin is Euclidean. For a vector $v$ with components $(v_x, v_y)$, let $$ ||v||^2 = f(v_x^2 + v_y^2) v_x^2 + v_y^2 = f(v \cdot v) v_x^2 + v_y^2 $$ where the $\cdot$ is the ordinary Euclidean dot product, and $f: \mathbb{R} \to \mathbb{R}$ is some nice function with $$ \lim_{r \to 0^+} f(r) = 1\\ \lim_{r \to \infty} f(r) = -1 $$
Examples of such an $f$ include $$ 2e^{-r^2} - 1 $$ and $$ \frac{1-r}{r+1} $$
Of course it no longer makes sense to call this a metric as it's not a symmetric bilinear form. I guess we could define $$ \left<u, v\right> = f(u \cdot v) u_xv_x + u_yv_y $$ but the notation would not be appropriate.
Does anyone know if such signature-changing spaces have a name? Where can I learn more about them?