Well-behaved metric What is a well-behaved metric in general relativity (GR)? Should every metric be well-behaved in GR? And what is the mathematical description for this kind of metric and what does it mean physically when we say a metric is well-behaved?
 A: "Well-behaved" is common Physics jargon for "this object should satisfy a number of conditions that I either don't want to list one by one or I don't remember right now". It is not a completely well-defined concept, but rather something like a Physics slang.
As mentioned in the comments, when asking for a well-behaved metric one probably will have in mind that it should be symmetric, invertible, and at least have well-defined second-derivatives, since these are necessary hypothesis to work with GR. However, one could or not be assuming extra things depending on context. For example, one could say "For a spacetime with a well-behaved metric and such and such conditions, this property holds" and mean that the metric must be analytic, which is far stronger than having well-defined second-derivatives.
As an example outside of GR, let us consider Fourier series. I've heard many physicists say that every well-behaved periodic function can be written as a Fourier series. This means that not all periodic functions can be written as a Fourier series (you can even find continuous functions that can't), but many periodic functions you'll encounter in Physics will satisfy the requirements to be written as a Fourier series. Of course, the term could also be used in a different way, such as "every well-behaved function can be written as a Fourier series", which leaves the periodicity hidden. It is still a necessary assumption, but the speaker chooses not to mention it either to avoid giving too much detail or for some other reason.
In short, it's pretty much a Physics slang to avoid listing all of the mathematical conditions that go into some result.
