Riemannian geometry

Riemannian geometry is a branch of differential geometry that looks at Riemannian manifolds and smooth manifolds that have a Riemannian metric. This means each space has a way to measure lengths and angles using an inner product defined on the tangent space. This inner product changes in a smooth way from one point to another.

Introduction

It is a type of non-Euclidean geometry. It breaks the parallel postulate and works with spaces that are curved instead of flat.

Classical theorems

Gauss–Bonnet theorem. The total of the Gaussian curvature in a compact two-dimensional Riemannian manifold is equal to $2\pi \chi (M)$, where $\chi (M)$ means the Euler characteristic.

Riemann believed that parallel lines always come together, because there is no way straight lines exist in a curved universe.

See also

Elliptic geometry

Geodesic