Riemannian geometry

In mathematics, Riemannian geometry has at least two meanings, one which described in this article and an other 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 amount of calculations. The following articles might serve as a rough introduction

1. Metric tensor
2. Riemannian manifold
3. Levi-Civita connection
4. Curvature
5. Curvature tensor.

The following articles might be also useful:

1. List of differential geometry topics
2. Glossary of Riemannian and metric geometry

Classical theorems in Riemannian geometry

What follows is a non complete 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 Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M), here χ(M) denotes the Euler characteristic of M.
2. Nash embedding theorems also called Fundamental Theorem 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) and get some information on the global structure of the space either some information on topologycal type of manifold or on behavior of point on "big" 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   and diameter  .
3. Gromov's almost flat manifolds. There is an ε_n > 0 such that if n-dimensional Riemannian manifold has a metric with sectional curvature   and diameter   then its finite cover diffeomorphic to a nil manifold.

Positive curvature

Positive sectional curvature

1. Soul Theorem. if M is 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 sum of its Betti numbers is at most C.

Positive Ricci curvature

1. Mayer's Theorem. If a compact Riemannian manifold has positive Ricci curvature then it has finite fundamental group.
2. Splitting theorem. If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic which minimize distance on each interval) then it is isometric to a direct product of the real line and a complete (n-1)-dimensional Riemannian manifold has nonnegative Ricci curvature
3. Bishop's inequality. the volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold manifold with positive Ricci curvature is at most as large as the volume of ball of the same radius r in Euclidean space.
4. Gromov Compactness Theorem. The set of all Riemannian manifolds which with positive Ricci curvature and diameter at most D is pre-compact in Gromov-Hausdorff metric.

Scalar curvature

1. n-dimensional torus does not admit a metric with positive scalar curvature.
2. If injectivity radius of a compact n-dimensional Riemannian manifold is   then average scalar curvature is at most n(n-1).

Negative curvature

Negative sectional curvature

1. For any two points of complete simply connected Riemannian manifold with nonpositive sectional curvature are joint by unique geodesic.
2. If M is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of its fundamental group of M is isomorphic to Z.

Negative Ricci curvature

1. Any compact Riemannian manifold with negative Ricci curvature has discrete isometry group.
2. Any smooth manifold admits a Riemannian metric with negative Ricci curvature.

External links

• MathWorld: Riemannian Geometry