Riemann was one of the first mathematicians to explore non-Euclidian geometry, along with Gauss (of course Gauss, he did everything), Bolyai and Lobachevsky.
One of Euclid’s postulates had troubled mathematicians for years. Postulates and axioms are the assumptions you make in order to start setting up any system in maths and back in Euclid’s time they were thought to be self-evidently true statements about the world (mathematicians have become more sophisticated about axiomatic systems now). Euclid’s 5th postulate didn’t seem quite as self-evident as it should have been. It didn’t seem axiomatic enough.
There are a number of ways of stating the 5th postulate. The commonest is to regard it as a statement about parallel lines, which says that if you have a straight line and a point lying outside that line you can draw one and only one line that passes through the point that is parallel to the line.
Most of us mightn’t see the difficulty. Surely it’s obvious that there’s only one line parallel to the other that passes through the point? But mathematicians didn’t like having to accept statements about what would or wouldn’t happen at infinity as axiomatic. They thought it wasn’t a firm enough foundation for geometry.
Some tried to derive the 5th postulate from the rest of Euclid’s geometry — with no luck. So a few mathematicians, in an attempt to provide a reductio ad absurdum proof, assumed an alternative and tried to derive a contradiction.
There are two obvious alternatives to Euclid. You could say that there are any amount of parallel lines; or there are no parallel lines at all. There are other possibilities; you could try the suggestion that there are precisely 17 parallel lines if you wished.
The suprising thing with this approach was that they couldn’t find a contradiction. They explored alternative geometries, uncovering all sorts of oddities (sum of the angles of a triangle always greater than 180°, or sum always less than 180°, for example). But they didn’t find contradictions. It became clear that these geometries were every bit as coherent and internally consistent as the familiar Euclidean geometry.
This belief that non-Euclidean geometries are just as consistent as Euclid isn’t simply an enlightened hunch. Although it isn’t possible to prove any axiomatic system to be absolutely consistent (that is, to be sure that we might not turn up a contradiction one day) it is possible to derive a proof that an axiomatic system is relatively consistent, which means that it is consistent if some other axiomatic system is consistent. For example, it’s been shown that if Euclidean geometry doesn’t have a contradiction buried in it somewhere then neither do the non-Euclidean geometries.
Riemann’s geometry held that no parallel lines could be drawn and that the sum of the internal angles of a triangle are always greater than 180°. It sounds peculiar but it’s understandable as a description of curved space. For example, imagine drawing a triangle on the surface of the Earth with an apex at the North Pole and the other two vertices on the equator. Two sides will drop from the Pole down to the equator - they’re lines of longitude and meet the equator at an angle of 90°, so that’s 180° so far: then there’s the angle the longitude lines meet at the North Pole. That’s something and it makes the interior angles of the triangle greater than 180°
Riemann’s geometry was later used by Einstein to describe the geometry of space — that’s one of the senses in which space can be described as being curved — the maths used to describe it can be thought of as the maths used to describe curved surfaces.
