Riemann Geometry: Curved Spaces and Einstein’s Equations

Riemann geometry studies spaces that can be curved instead of flat.
It extends Euclidean geometry to describe shapes like spheres and saddles.
The key idea is a metric, which defines distance in curved space.
Geodesics are the “straightest” possible paths in curved geometry.
Einstein used Riemann geometry to show that gravity is the curvature of spacetime.
Massive objects like stars and planets bend spacetime, guiding motion along geodesics.
Black holes are extreme examples of spacetime curvature described by Riemann geometry.
The Riemann curvature tensor encodes how much and in what way space is curved.
GPS satellites must account for spacetime curvature to give accurate positions.
At its core, Riemann geometry is the mathematics that turned gravity into geometry.