Riemannian geometry

This article should be merged with Riemannian manifold

Riemannian geometry is a description of an important family of geometries, first put forward in generality by Bernhard Riemann in the nineteenth century. It is an intrinsic description, of what is now called a Riemannian manifold. 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 axiomatic footing, as are a broad range of geometries whose metric properties vary from point to point.

The characteristic structure in Riemannian geometry is a metric tensor defined on the tangent space, from point to point. This gives a local idea of angle, length and volume. From these global quantities can be derived, by integrating local contributions.

Table of contents

1 Brief on the metric tensor
2 Exponential map
3 Connections and Curvature
4 Killing fields
5 Jacobi fields
6 Orthonormal frames
7 External Links

Brief on the metric tensor

The metric tensor g is a twice differentiable (at the very least) symmetric pointwise linear map of two sections of the tangent bundle, X and Y, g(X,Y) which is positive definite (i.e. g(X,X)>= 0 with equality only when X=0 ).

The metric tensor, conventionally notated as , as a 2-dimensional tensor (making it a matrix), that is used to measure distance in a coordinate space or manifold. is conventionally used to notate the components of the metric tensor. (The elements of the matrix)

The length of a segment of a curve parameterized by t, from a to b, is defined as:

This is sometimes written as where here, the Einstein summation notation is used.

For example, given a two-dimensional Euclidean metric tensor:

The length of a curve reduces to the familiar Calculus formula:

The extremals of this functional are called geodesics.

The area or volume(or whatever the term is for the number of dimensions you are working with) is given by the formula:

where g is the determinant of the metric. For instance in polar coordinates the metric is:

Its determinant is r², thus the area is given by:

Exponential map

On a Riemannian manifold there is an exponential map, defined in each point, which maps the tangent space to the manifold itself. It is obtained by moving along the geodesic starting at this point, and going in given direction to the corresponding distance along the geodesic. The injectivity radius is the maximal radius of a ball such that this map is still a bijection.

Connections and Curvature

Given a metric tensor, there is a unique torsion free connection which preserves this metric. It is called the Levi-Civita connection. The Riemanian curvature is the curvature of this connection. Its study is very important for understanding of Riemanian geometry.

Killing fields

Killing fields are vector fields which have the property that they preserve the Riemannian metric g. In other words, the flow diffeomorphisms act as isometries.

Killing field is determined uniquely by a vector at some point and its gradient. Killing fields form a Lie algebra of dimension not greater than ((n+1)n)/2.

Killing fields generate continuous group of isometries of a manifold. Many manifolds have only a discrete group of isometries, and support no Killing fields.