Riemannian geometry is a kind of non-Euclidean geometry, and parallel lines in non-Euclidean geometry can also intersect. The geometry we usually learn is Euclidean geometry, which is based on the five * * * proposed by Euclid. And the fifth * * * can't prove it with facts. So there is non-Euclidean geometry.

A basic law in Riemannian geometry is that any two straight lines on the same plane have common points (intersections). In Riemannian geometry, the existence of parallel lines is not recognized, and its other postulate says that straight lines can extend indefinitely, but the total length is limited. The model of Riemannian geometry is a sphere that has been properly "improved".

Euclidean geometry, Roche geometry and Riemann geometry are three different geometries. All the propositions of these three kinds of geometry constitute a strict axiom system, which meets the requirements of harmony, completeness and independence. So these three geometries are all right.

Extended data

The scope of application of Euclidean geometry and non-Euclidean geometry

Euclidean geometry mainly studies the geometry of plane structure and solid geometry, while non-Euclidean geometry studies irregular surfaces. Euclidean geometry can be used to study the geometry on the plane, that is, plane geometry.

The study of Euclidean geometry in three-dimensional space is usually called solid geometry. Non-Euclidean geometry is suitable for studying abstract space, that is, more general space form, which makes the development of geometry enter a brand-new stage characterized by abstraction. Non-Euclidean geometry also applies to Einstein's general theory of relativity.