People are aware of Euclid as the "Father of Geometry", every fundamental principle of Geometry whether it is the sum of angles or some defined shape, Euclid developed the foundation for all of them. His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. But the thing about Bernhard Riemann is that you may never hear of him if you are not doing a course on the theory of relativity (especially General Theory of relativity). If you know about electricity and magnetism then you might have known about Carl Freidrich Gauss. Gauss was a genius mathematician, in physics we used his methods of calculating flux through a particular region. Riemann was Gauss's favorite and brilliant student, just like his mentor Riemann developed a form of mathematics that revolutionized the way of physics. He developed a new form of the geometry of curved spaces which later on known Riemannian Geometry. This helps Einstein to work on his ideas about the curved spacetime. As we know in Science intuition itself cannot fulfill the notion of principles, Einstein needs that mathematics. We can easily say that without Riemann Einstein couldn't make through the development of his famous field equations.
Okay enough history now let's see what is Riemannian Geometry. There is a very fundamental difference in the Euclidean and Riemannian geometry which is that one works on flat space and the other works on curved. we can say that the Riemannian is a more generalized case because if we out flat space condition Riemann then it will reduce into Euclid. The curved space here means that it should be locally euclidean and globally curved. Our earth is a perfect example of this statement as we can see that our dimension is lower than that of the earth so for us (locally) it seems flat. But for real (globally) we know that it is a sphere (ellipsoid to be precise). Now here the laws don't seem to follow that clearly. We know that the sum of the angles of a triangle is 180 degrees. The statement is correct no doubt but the idea is restricted to the flat space. If you draw a triangle on a ball you will see that it seems more stretchy. It is because now the sum of the angles is not 180 it is more than 180. We call this as positive curvature. Where if you draw the triangle inside a hemispherical ball. Then the sum will be less than 180. We call it a negative curvature.
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