What is Riemannian Geometry?

Euclidean Geometry is the study of flat space. Between every pair of points there is a unique line segment which is the shortest curve between those two points. These line segments can be extended to lines. Lines are infinitely long in both directions and for every pair of points on the line, the segment of the line between them is the shortest curve that can be drawn between them. Furthermore, if you have a line and a point which isn't on the line, there is a second line running through the point, which is parallel to the first line (never hits it). All of these ideas can be described by drawing on a flat piece of paper. From the laws of Euclidean Geometry, we get the famous theorems like Pythagorus' Theorem and all the formulas you learn in trigonometry, like the law of cosines. In geometry you also learned how to find the circumference and area of a circle.

Euclidean Geometry is the study of flat space. Between every pair of points there is a unique line segment which is the shortest curve between those two points. These line segments can be extended to lines. Lines are infinitely long in both directions and for every pair of points on the line, the segment of the line between them is the shortest curve that can be drawn between them. Furthermore, if you have a line and a point which isn't on the line, there is a second line running through the point, which is parallel to the first line (never hits it). All of these ideas can be described by draw