What is the mathematical name of the "curved space" in Einstein's theory?

Question

Newtonian mechanics assumed a Euclidian geometry.

Einsteinian mechanics is based on non-Euclidian geometry.

However, there are many types of non-Euclidian geometry, so what is the mathematical name of the geometry used in Einstein's relativity theory?

Answer 1

Riemannian geometry is the backbone of Einstein's General Theory of Relativity.

To quote, in his own words:

"[In 1912] I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss's theory was contained in the geometry course given by Geiser when I was a student... I realized that the foundations of geometry have physical significance. My dear friend the mathematician Grossmann was there when I returned from Prague to Zürich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann's theory , namely, whether the invariants of the line element could completely determine the quantities I had been looking for."

Albert Einstein, as quoted by Abraham Pais in Subtle is the Lord, Pais's scientific biography of Einstein.