Riemann homogeneous space

In the mathematical field of differential geometry , a Riemann homogeneous space (often just a homogeneous space ) is a space that "looks the same in all points".

definition

A Riemann homogeneous space is a Riemann manifold whose isometric group acts transitive , i.e. H. for every two points there is an isometry with . ${\ displaystyle M}$ ${\ displaystyle x, y \ in M}$ ${\ displaystyle g \ in \ operatorname {isom} (M)}$ ${\ displaystyle g (x) = y}$

Description using Lie groups

Every Riemannian homogeneous space is of the form

{\ displaystyle M = G / H}

for a Lie group and a compact subgroup . ${\ displaystyle G}$ ${\ displaystyle H \ subset G}$

Group Lie Conversely, for a and a completed subassembly of the quotient space a Hausdorff differentiable manifold and each of the adjoint effect of on the Lie algebra invariant scalar defines a left-invariant Riemannian metric, with a Riemannian homogeneous space. Such an -invariant scalar product on exists if and only if is compact. ${\ displaystyle G}$ ${\ displaystyle H}$ ${\ displaystyle G / H}$ ${\ displaystyle H}$ ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle G / H}$ ${\ displaystyle Ad (H)}$ ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle H}$

Riemannian metric

By definition, a Riemann homogeneous space has an -invariant metric, which can be elevated to a left-invariant metric . With regard to these metrics, the quotient mapping is a Riemannian submersion . In particular, one can calculate the curvature of with the O'Neill formula if one knows the curvature of . ${\ displaystyle G / H}$ ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle G \ to G / H}$ ${\ displaystyle G / H}$ ${\ displaystyle G}$

Examples

Every Lie group with a left-invariant metric is a Riemann homogeneous space.
Every symmetrical space is a Riemannian homogeneous space.
There are non-Riemannian homogeneous spaces with a non-compact subgroup . ${\ displaystyle G / H}$ ${\ displaystyle H \ subset G}$

literature

Jeff Cheeger, David G. Ebin: Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.