Myers-Steenrod's theorem

This article deals with Myers-Steenrod's theorem on isometric groups. For the Myers-Steenrod theorem about distance-maintaining maps see Isometry (Riemannian Geometry) # Myers-Steenrod's Theorem .

The set of Myers-Steenrod is a theorem from the mathematical field of differential geometry .

It says that the isometric group of every complete Riemannian manifold is a Lie group .

The phrase is from Norman Steenrod and Sumner Byron Myers .

Examples

The isometric group of the unit sphere is the orthogonal group . ${\displaystyle S^{n}}$ ${\displaystyle O(n+1)}$

The isometric group of the hyperbolic plane is the projective linear group . The isometric group of 3-dimensional hyperbolic space is . ${\displaystyle PGL(2,\mathbb {R} )}$ ${\displaystyle PGL(2,\mathbb {C} )}$

Proof idea

In a connected Riemannian manifold choose a point and its exponential map . The images of the 1-dimensional subspaces in under the exponential mapping are exactly the geodesics through . From the completeness of, it follows with the Hopf-Rinow theorem that every point in lies on such a geodesic . ${\displaystyle M}$ ${\displaystyle x}$ ${\displaystyle \exp _{x}\colon T_{x}M\to M}$ ${\displaystyle T_{x}M}$ ${\displaystyle x}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle x}$

Now choose linearly independent vectors in and designate their image points with . An isometry maps geodesics in geodesics and it follows from the above that an isometry is already clearly defined by the images of . ${\displaystyle n=\dim(M)}$ ${\displaystyle T_{x}M}$ ${\displaystyle p_{1},\ldots ,p_{n}}$ ${\displaystyle \exp _{x}}$ ${\displaystyle x,p_{1},\ldots ,p_{n}}$

So we get an embedding of the isometric group in the product of copies of the manifold . One can show that the image of this embedding is a differentiable submanifold and that the group operations in this manifold structure are differentiable. This becomes a Lie group. ${\displaystyle \operatorname {Isom} (M)}$ ${\displaystyle n+1}$ ${\displaystyle M}$ ${\displaystyle \operatorname {Isom} (M)}$

generalization

More generally, the isometric group of a room is always a Lie group. -Spaces are a class of metric measure spaces that contain all Riemannian manifolds of the dimension with Ricci curvature and are closed under the Gromov-Hausdorff convergence of metric measure spaces. ${\displaystyle RCD^{*}(K,n)}$ ${\displaystyle RCD^{*}(K,n)}$ ${\displaystyle \leq n}$ ${\displaystyle Ric\geq K}$

literature

• SB Myers, NE Steenrod: The group of isometries of a Riemannian manifold. Ann. of Math. (2) 40 (1939), no. 2, 400-416.

Individual evidence

1. ↑ L. Guijarro, J. Santos-Rodríguez: On the isometry groups of RCD * (K, N) -spaces , manuscripta mathematica 158, 441–461 (2018)
2. ↑ G. Sosa: The isometry group of an RCD * -space is Lie , Potential Analysis 49, 267-286 (2018)