👉 Riemannian geometry, or Riemannian mathematics, is a branch of differential geometry that studies geometric objects with a metric that varies smoothly from point to point. It generalizes Euclidean geometry by allowing for curved spaces, where distances and angles are defined using a Riemannian metric—a tensor that assigns inner products to tangent spaces at each point. This framework is crucial for understanding manifolds, which are higher-dimensional analogs of curves and surfaces, and it plays a pivotal role in physics, particularly in general relativity, where spacetime is modeled as a curved Riemannian manifold. Key concepts include geodesics (the shortest paths in curved spaces), curvature tensors, and the Riemann curvature tensor, which quantify how much a manifold deviates from being flat.
