What is Minkowski's geometric when talking about relativity?
Maybe you mean metric.
The metric in relativity is what allows us to measure distances between two events in spacetime. The Minkowski metric is:
[tex](ds)^2 = (c\,dt)^2 - (dx)^2 - (dy)^2 - (dz)^2[/tex],
possibly with reversed signs on the right hand side (different people define it differently).
ds is the spacetime interval, c is the speed of light, dt is the time interval between the two events, and dx, dy and dz are the spatial intervals in the three spatial directions.
The important thing is that ds is the same for two events regardless of which reference frame you are using, whereas dt, dx, dy and dz change when you change reference frames.
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