Hello Gianluca, I have skimmed my text again and noticed that I did not thank you at all. Sorry. I would like to make up for that: Thanks for the helpful tip. - I am a retired physicist, interested in general relativity. I wish you all the best and remain with best regards Bernd

One way to get an informative picture is with ContourPlot:

Show[ContourPlot[x^2 - y^2, {x, -2, 2}, {y, -2, 2}, Contours -> 20, 
  PlotPoints -> 50, ContourShading -> False, ContourStyle -> Black],
 ContourPlot[x y, {x, -2, 2}, {y, -2, 2}, Contours -> 20, 
  PlotPoints -> 50, ContourShading -> False, ContourStyle -> Red]]

Disappointingly, this concise version shows only the 0-level contours:

ContourPlot[{x^2 - y^2, x y}, {x, -2, 2}, {y, -2, 2},
 Contours -> 20, ContourShading -> False]