How do we explains space-time through Riemann Calculus?
Riemann calculus is really needed only to describe curvature, not space-time.
The mathematical structure of space-time, like the mathematical structure of just space, is that of a manifold. A manifold is built on top of some other math concepts, I'm not sure if I should go into that level of detail, for the moment I won't.
We had another thread recently on why we combine space and time into a unified "space-time". This basically comes not from mathematics, but rather from special relativity.
You might want to read this: https://www.physicsforums.com/threads/very-little-excuse-to-ask-a-question-cold.765735/
Essentially, your question is "please write a textbook on GR for me here". That's too much to ask, and doing so disrespectful of other people's time. You need to start doing some work on your own.
What Vanadium50 said. This thread is closed, but more specific and focused questions in new threads will be welcome.
Separate names with a comma.