Riemannian Geometry Classical Theorems In Riemannian Geometry

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 geometryHyperbolic geometry also called saddle geometry or Lobachevskian geometry is the non-Euclidean geometry obtained by replacing the parallel postulate with the hyperbolic postulate which states: "Given a line L and any point A not on L at least two distinct) of Non-Euclidean geometryThe term non-Euclidean geometry (also spelled: non-Euclidian geometry describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of par, as well as Euclidean geometryEuclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry someti 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 manifoldIn differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, (0,2) tensor which is nondegenerate at each point on the manifold. This tensor is called a pseudo-Riemannian metric or, simply, a pseudo-) metris, which (in dimension four) are the main objects of general relativityGeneral relativity (GR or general relativity theory (GRT is the theory of gravitation published by Albert Einstein in 1915. The conceptual core of general relativity, from which its other consequences largely follow, is the Principle of Equivalence which 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:

1 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.

1.1 General theorems

Gauss-Bonnet Theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to where denotes the Euler characteristic of M.
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.

1.2 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.

1.2.1 Pinched sectional curvature

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.
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 .
Gromov's almost flat manifolds. There is an such that if an n-dimensional Riemannian manifold has a metric with sectional curvature and diameter then its finite cover is diffeomorphic to a nil manifold.

1.2.2 Positive curvature

1.2.2.1 Positive sectional curvature

Soul theorem. if M is a non-compact complete positively curved n-dimensional Riemannian manifold then it is diffeomorphic to Rⁿ.
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.

1.2.2.2 Positive Ricci curvature

Myers theorem. If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
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
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.
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.

1.2.2.3 Scalar curvature

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

1.2.3 Negative curvature

1.2.3.1 Negative sectional curvature

Any two points of a complete simply connected Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
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.

1.2.3.2 Negative Ricci curvature

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

2 External links

MathWorld: Riemannian Geometry

Riemannian geometry

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