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.http://mathworld.wolfram.com/RiemannianGeometry.htmlhttps://d-nb.info/gnd/4128462-8https://thes.bncf.firenze.sbn.it/termine.php?id=31545
 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 , where  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

 References 

Category:Geometry