# Riemannian geometry

In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry.

In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i.e. a choice of positive-definite quadratic form on a manifold's tangent spaces which varies smoothly from point to point. This gives in particular local ideas of angle, length of curves, and volume. From those some other global quantities can be derived, by integrating local contributions.

It was first put forward in generality by Bernhard Riemann in the nineteenth century. As particular special cases there occur the two standard types (spherical geometry and hyperbolic geometry) of Non-Euclidean geometry, as well as Euclidean geometry itself. These are all treated on the same basis, as are a broad range of geometries whose metric properties vary from point to point.

Any smooth manifold admits a Riemannian metric and this additional structure 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 dimension four) are the main objects of general relativity theory.

There is no easy introduction to Riemannian geometry. One should work quite a while to build some geometric intuition here; it is usually done by doing enormous amounts of calculations. The following articles might serve as a rough introduction:

The following articles might be also useful:

## Classical theorems in Riemannian geometry

What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance, beauty, and simplicity of formulation.

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

1. Gauss-Bonnet Theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to $\displaystyle 2\pi\chi(M)$ where $\displaystyle \chi(M)$ denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.
2. Nash embedding theorems also called Fundamental Theorems of Riemannian geometry. They state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn.

### Local to global theorems

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

1. 1/4-pinched Sphere Theorem. If M is a complete n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1 and 4 then M is homeomorphic to n-sphere.
2. Cheeger's Finiteness theorem. Given constants C and D there are only finitely many (up to diffeomorphism) compact n-dimensional Riemannian manifolds with sectional curvature $\displaystyle |K|\le C$ and diameter $\displaystyle \le D$ .
3. Gromov's almost flat manifolds. There is an $\displaystyle \epsilon_n>0$ such that if an n-dimensional Riemannian manifold has a metric with sectional curvature $\displaystyle |K|\le \epsilon_n$ and diameter $\displaystyle \le 1$ then its finite cover is diffeomorphic to a nil manifold.

#### Positive curvature

##### Positive sectional curvature
1. Soul theorem. if M is a non-compact complete positively curved n-dimensional Riemannian manifold then it is diffeomorphic to Rn.
2. Gromov's Betti number theorem. There is a constant C=C(n) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then the sum of its Betti numbers is at most C.
##### Positive Ricci curvature
1. Myers theorem. If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
2. Splitting theorem. If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic which minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete (n-1)-dimensional Riemannian manifold which has nonnegative Ricci curvature
3. Bishop's inequality. the volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space.
4. Gromov's compactness theorem. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric.
##### Scalar curvature
1. The n-dimensional torus does not admit a metric with positive scalar curvature.
2. If the injectivity radius of a compact n-dimensional Riemannian manifold is $\displaystyle \ge \pi$ then the average scalar curvature is at most n(n-1).

#### Negative curvature

##### Negative sectional curvature
1. Any two points of a complete simply connected Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
2. If M is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of the fundamental group of M is isomorphic to Z.
##### Negative Ricci curvature
1. Any compact Riemannian manifold with negative Ricci curvature has a discrete isometry group.
2. Any smooth manifold admits a Riemannian metric with negative Ricci curvature.