<section begin="Lede" />**Riemannian geometry** is the branch of [differential geometry](https://bliptext.com/articles/differential-geometry) that studies [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold), defined as [smooth manifolds](https://bliptext.com/articles/manifold) with a Riemannian metric (an [inner product](https://bliptext.com/articles/inner-product) on the [tangent space](https://bliptext.com/articles/tangent-space) at each point that varies [smoothly](https://bliptext.com/articles/smooth-function) from point to point). This gives, in particular, local notions of [angle](https://bliptext.com/articles/angle), [length of curves](https://bliptext.com/articles/arc-length), [surface area](https://bliptext.com/articles/surface-area) and [volume](https://bliptext.com/articles/volume). From those, some other global quantities can be derived by [integrating](https://bliptext.com/articles/integral) local contributions. Riemannian geometry originated with the vision of [Bernhard Riemann](https://bliptext.com/articles/bernhard-riemann) expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the [differential geometry of surfaces](https://bliptext.com/articles/differential-geometry-of-surfaces) in **R**3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of [geodesics](https://bliptext.com/articles/geodesic) on them, with techniques that can be applied to the study of [differentiable manifolds](https://bliptext.com/articles/differentiable-manifold) of higher dimensions. It enabled the formulation of [Einstein](https://bliptext.com/articles/albert-einstein)'s [general theory of relativity](https://bliptext.com/articles/general-theory-of-relativity), made profound impact on [group theory](https://bliptext.com/articles/group-theory) and [representation theory](https://bliptext.com/articles/representation-theory), as well as [analysis](https://bliptext.com/articles/global-analytic-function), and spurred the development of [algebraic](https://bliptext.com/articles/algebraic-topology) and [differential topology](https://bliptext.com/articles/differential-topology).<section end="Lede" />

Introduction

Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry. Every smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature. The following articles provide some useful introductory material:

Classical theorems

What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below). The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.

General theorems

Geometry in large

In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.

Pinched sectional curvature

Sectional curvature bounded below

Sectional curvature bounded above

Ricci curvature bounded below

Negative Ricci curvature

Positive scalar curvature