Riemannian geometry
The Riemannian geometry is a branch of differential geometry and was named after Bernhard Riemann. In this theory, the geometric properties of a Riemannian manifold can be examined. These are smooth manifolds with a kind of scalar product. With this function you can measure angles, lengths, distances and volumes.
Formation
The first works of differential geometry go back to Carl Friedrich Gauss. He founded the theory of curved surfaces that were embedded in three-dimensional space. The Riemannian geometry received its decisive impetus in 1854 in Riemann's habilitation lecture entitled " On the Hypotheses which lie the underlying geometry ." In this work he introduced the Riemannian metrics, which were later named after him. In contrast to Gaussian he considered not only space, but higher-dimensional curved spaces. These spaces, however, were still embedded in a Euclidean space. The abstract topological definition of differentiable and thus in particular of Riemannian manifolds was first developed in the 1930s by Hassler Whitney. Particularly well known is the statement that any differentiable manifold can be embedded. This result is now known as the embedding theorem of Whitney.
Riemann's ideas were further developed in the second half of the 19th century by Elwin Bruno Christoffel ( covariant derivative, Christoffel symbols) and in the context of Tensorkalküls by Gregorio Ricci and Tullio Levi -Civita Curbastro.
Buoyancy was the theory by the general theory of relativity by Albert Einstein ( 1916), the basis of which are the pseudo- Riemannian manifolds. In this context, the theory in particular Hermann Weyl and Élie Cartan was developed that proved the role of affine connections and parallel transport.
Important objects and statements
The central object of the Riemannian geometry is the Riemannian manifold. This is a smooth manifold together with a map defining an inner product of the tangent space at each point, that is a positive definite, symmetric bilinear form
With this Riemannian metric is obtained as in conventional vector spaces with scalar product, the terms of the arc length, the distance and angle.
A mapping between Riemannian manifolds which receives the Riemannian metric ( and hence the length and angle of tangent and the length of curves), ie Riemannian isometry. However, as an illustration does not need the distance between points to obtain and is therefore generally not isometric in the sense of metric spaces.
Another induced by the Riemannian metric object is the Riemannian volume form. This makes it possible to measure volume on manifolds, and is therefore a central component of the integration theory on oriented Riemannian manifolds.
As to the (continuous ) Riemannian manifolds, a distance is defined, you can also transfer the concept of completeness. The set of Hopf Rinow is central. He says, among other things, that the generalized ( geodesic ) completeness is on the manifold is equivalent to the completeness as a metric space. Another important statement is the embedding theorem of Nash. Similar to the embedding theorem of Whitney, he says that one can embed any Riemannian manifold into a sufficiently large dimension. However, compared to the embedding theorem of Whitney, he makes a stronger statement, for he goes on to say that the embedding receives lengths and angles. Embedding means here that the manifold as a subset of can be understood.
In addition to the metric properties we are interested in the Riemannian geometry for curvature quantities. In the theory of surfaces the Gaußkrümmung was examined before Riemann's work. In higher-dimensional manifolds to study the curvature is more complex. For this purpose, the Riemannian curvature tensor has been introduced. With the help of this tensor we define the sectional curvature, it can be understood as a generalization of Gaußkrümmung and is the main notion of curvature of Riemannian geometry is used in the comparison theory application in particular. Linear correlations on vector bundles also play an important role in the theory of curvature, in particular even for the definition of the Riemannian curvature tensor. On Riemannian manifolds there is a clear linear relationship, which is torsion-free and compatible with the Riemannian metric. This statement is often referred to as the law of Riemannian geometry. The corresponding connection is called the Levi- Civita connection.
Comparison theory
In Riemannian geometry, there are some statements that are referred to traditionally as a comparative sentences. These statements are examined, for example, Riemannian manifolds whose sectional curvature or Ricci curvature is bounded from above or below. For example, the set of Bonnet makes a statement about manifolds whose sectional curvature is bounded by a positive number down. A stronger statement is the set of Myers, who derives the same statement from the weaker condition of limited by a positive number down Ricci curvature. The set of Cartan -Hadamard, however, shows a link between manifolds with nonnegative sectional curvature and their universal overlay space. One of the most important sets of comparisons in Riemannian geometry is the set of spheres. This means that compact, simply connected Riemannian manifolds, the inequality holds for the sectional curvature, homeomorphic to the sphere are.