Isometry (Riemannian geometry)

In differential geometry , a branch of mathematics , images are called local isometrics if they contain the Riemannian metric . As isometrics are called diffeomorphisms , the local isometrics are.

A local isometry maps curves to curves of the same length, but it does not necessarily have to contain distances. (For example, a Riemann superposition is a local isometric drawing.) An isometric drawing is also given distances. An isometry in the sense of Riemannian geometry is always an isometry between the metric spaces. Conversely, according to a theorem of Myers-Steenrod, every mapping between Riemannian manifolds that preserves distances is an isometry in the sense of Riemannian geometry.

definition

Let and be two Riemannian manifolds . A diffeomorphism is an isometry if ${\displaystyle (M,g)}$ ${\displaystyle (M',g')}$ ${\displaystyle f\colon M\to M'}$

${\displaystyle g=f^{*}g'}$

holds, where denotes the pullback of the metric tensor . So it's supposed to be the equation ${\displaystyle f^{*}g'(v,w):=g'\left(f_{*}v,f_{*}w\right)}$

${\displaystyle g(v,w)=g'\left(f_{*}v,f_{*}w\right)}$

apply to all tangential vectors in , where denotes the push forward of . ${\displaystyle v,\,w}$ ${\displaystyle T_{m}M}$ ${\displaystyle f_{*}}$ ${\displaystyle f}$

A local isometry is a local diffeomorphism with . ${\displaystyle g=f^{*}g'}$

example

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

Myers-Steenrod's theorem

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

Isometry group

The isometrics of a metric space always form a group. Steenrod and Myers proved in 1939 that the isometric group of a Riemann manifold is always a Lie group ( Myers-Steenrod theorem ).

Examples:

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

The dimension of the isometric group of an n-dimensional compact manifold is at most . ${\displaystyle {\tfrac {1}{2}}n(n+1)}$

swell

1. ^ SB Myers, NE Steenrod: The group of isometries of a Riemannian manifold. In: Ann. of Math. 2, No. 40, 1939, pp. 400-416.
2. ^ RS Palais: On the differentiability of isometries. In: Proceedings of the American Mathematical Society. 8, 1957, pp. 805-807.