ricardo_gt,
You mentioned “gearboxes” and lathe in your first post.
If you are investigating gear manufacturing specifically (not the entire transmission or “gearbox”), then about the only thing a lathe will get used for is making blanks.
Gear cutting methods are very mature, at least when it comes to involute based gear design (which probably covers 99% of the gears in existence). This would include spur, helical, bevel, spiral bevel and hypoid gears.
Hobbing and shaping would be the primary methods of spur and helical gear cutting (external gears) — and shaping (blind holes) and broaching (through holes) for internal gears. These “methods” (admittedly with machines modernized since the advent of CNC controls) remain fundamentally and geometrically unchanged for almost 100 years. Cutting tools and cutting oils have improved, so manufacturing cycles times have come down, however the general methods remain unchanged.
There are newer and possibly faster methods of gear cutting which can work well (under “certain” circumstances), namely: gear skiving and gear skudding.
And of course, external splines can often be “formed” via rolling… which is very fast, though does not lend itself well to full depth gears with pressure angles below 45°.
Large, specialty gears can be cut in a 5-axis CNC milling machine, though this is not a process which would be used in mass production (far too slow).
For highly accurate gears, grinding is a common “follow-up” in the complete manufacturing process — where a gear is typically cut slightly oversize in a “pre-grind” form, then heat treated, then ground. Once again, this is also a very developed process, (though once again has improved with the advent of CNC controls).
PM