I'm currently studying non-Euclidean geometry and recently learned that a sphere is in 3-dimensional Euclidean space, but its surface is not.
According to my findings, non-Euclidean spaces are "spaces where the parallel postulate does not hold." (History of manifolds and varieties – Wikipedia) meaning that if there was a line and we were to draw two lines such that the inner angles are less than 90 degrees, the two lines would never meet. Please correct me if I'm understanding this fundamentally incorrectly.
If I have understood it correctly, then does the surface of a sphere satisfy this postulate? If I were to draw a line on the surface of a sphere and draw two lines such that the inner angle they make with the first line is less than 90 degrees, I'm pretty sure that they would meet somewhere on the surface.
Would anyone be kind enough to help me understand this concept? Thanks in advance.
P.S., I've taken a look at this Math Stack Exchange question: Spherical Geometry and Playfair's Axiom but it didn't help so much.
Best Answer
One thing to keep in mind is that line segments on the surface of a sphere do not work the same way as the Euclidean metric in space. For instance, the shortest distance along the surface from Singapore to San Francisco passes through Tokyo, which is not what you would expect at all from looking at the Pacific Ocean on a rectangular map.
So lines on a sphere are "great circles", which represent circles in space whose center is the center of the sphere. Thinking of the globe again, all longitudinal lines are great circles, but the Equator is the only latitudinal line that is a great circle. Based on that definition of lines in the sphere, it is not hard to convince yourself that any two distinct lines on a sphere must intersect in two places.