Isometry (Riemannian geometry)

In differential geometry, a branch of mathematics called you pictures as local isometries, when they receive the Riemannian metric. As isometries are called diffeomorphisms, the local isometries are.

A local isometry from forming curves on curves of equal length, but it does not necessarily get distances. ( For example, a Riemannian overlay is a local isometry. ) An isometry also receives distances. An isometry in the sense of Riemannian geometry is thus always an isometry between metric spaces. Conversely, according to a set of Steenrod -Myers each spacing - preserving map between Riemannian manifolds is an isometry in the sense of Riemannian geometry.

Definition

Let and be two Riemannian manifolds. A diffeomorphism is an isometry if

, where the pullback of the metric tensor referred. So we want the equation

For all tangent vectors to apply.

A local isometry is a local diffeomorphism with.

Example

The isometries of Euclidean space are rotations, reflections and shifts.

Set of Steenrod -Myers

Norman Steenrod and Sumner Byron Myers proved in 1939 that any distances -preserving continuous map between connected Riemannian manifolds must be an isometry. ( In particular, such a mapping is always differentiable. ) A simpler proof was given by Richard Palais in 1957.

Isometry group

The isometries of a metric space always form a group. Steenrod and Myers proved in 1939 that the isometry group of a Riemannian manifold is always a Lie group.

Examples:

• The isometry group of the n-dimensional sphere is the orthogonal group O ( n 1).
• The isometry group of the n-dimensional hyperbolic space is the Lorentz group O ( n, 1).

The dimension of the isometry a n-dimensional manifold is most compact.

Swell

• Riemannian geometry