Riemannian manifold

A Riemannian manifold or Riemannian space is an object from the mathematical branch of Riemannian geometry. These manifolds have the additional property that they have a metric similar to a pre-Hilbert space. With this Riemannian metric then can describe the essential geometric properties of the manifold. To apply to any Riemannian manifold the following, partially equivalent, properties:

• The shortest distances between different points ( the so-called geodesics ) are not necessarily straight pieces, but can be curved corners.
• The angle sum of triangles, in contrast to the plane be greater ( eg ball ) or smaller ( hyperbolic space ) than 180 °.
• The parallel translation of tangent vectors along curves closed can change the direction of the vector.
• The result of a parallel shift of a tangent vector also depends on the way, is shifted along which the tangent vector.
• The curvature is generally a function of the location on the manifold.
• Distance measurements between different points are only possible with the help of a metric that can depend on the location on the manifold.

The more general notion of pseudo- Riemannian or semi- Riemannian manifold is in general relativity theory of crucial importance, since in this space-time is described as such.

• 2.1 Euclidean vector space
• 2.2 Induced metric

Mathematical definition

Riemannian manifold

A Riemannian manifold is a differentiable -dimensional manifold with a function which assigns to each point of a dot product of the tangent space, that is a positive definite, symmetric bilinear form

The differentiable of depends. That is, given differentiable vector fields is

A differentiable function. The function is called a Riemannian metric or metric tensor, but is not a metric in the sense of metric spaces.

Riemannian manifolds as metric spaces

The Riemannian metric is not a metric, but a scalar. You can win a metric However, similar to the theory of Skalarprodukträume from the scalar product. Since there is no need to be path on manifolds, the shortest path even, the definition is slightly more complicated. The distance function on a ( connected ) Riemannian manifold is then defined by

It goes through all the ( piecewise ) differentiable paths and connect, and denotes the length of which in accordance with

Is defined. The functional is also called the length functional. One way of realized locally ( ie for sufficiently closely spaced points), the shortest path is called geodesic.

This metric induced back to the original topology. Since one can show that every differentiable -dimensional manifold has Riemannian metrics can thus also show that any differentiable -dimensional manifold is metrizable. Similar to metric vector spaces, we can speak of complete Riemannian manifolds. The set of Hopf Rinow is the central result regarding the completeness Riemannian manifolds.

Examples

Euclidean vector space

A Euclidean vector space is isometrically isomorphic to the standard scalar

The vector space can be understood as a differentiable manifold and along with the standard scalar product it becomes a Riemannian manifold. In this case, the tangent space is identical to the output space, so again the.

Induced metric

Since the tangent bundle of a submanifold of a Riemannian manifold is a subset of the tangent bundle of the metric of also can be applied to the tangent to the submanifold. The resulting metric of the submanifold is therefore also called the induced metric. The submanifold together with the induced metric again a Riemannian manifold.

Induced metrics are used in particular in the geometrical study of curves and surfaces, as a real submanifolds of, use.

History

Gauss ' theory of curved surfaces using an extrinsic description, that is, the curved surfaces will be described with the aid of a surrounding, Euclidean space. On the other hand Riemann represents a more abstract approach. This approach and the associated definitions led Riemann in his habilitation lecture about the hypotheses which underlie geometry of 10 June 1854 the University of Göttingen. There are also many definitions were presented, which are still used today in modern mathematics. However, of paracompact spaces was not yet mentioned. Instead of curves and tangent Riemann used then infinitesimal line elements.

Since the beginning of the 19th century so-called non-Euclidean geometries are discussed. The Riemannian geometry has this to describe these geometries from a general point of view, just the appropriate definitions and the appropriate language. The notion of Riemannian manifold formed the beginning of the 20th century a fundamental starting point for the development of general relativity.