simple-symmetric-space

[symmetric-space-locally] := under geodesic coordinate, inversion is isometry. equivalently,

Example quadratic manifold, simple-Lie-group and related symmetric-space

constant-sectional-curvature ==> symmetric-space

simple-Lie-group := Lie-algebra & Lie-bracket cannot decompose

Killing form := for in the tangent space at

Then define the metric at to be the metric at other places generated by the action, and it is bi-invariant i.e. both forms of group action give the same metric

Such a definition makes the group action an isometry of the Killing-form

Killing-form can also be defined for non-simple-Lie-groups

Question Motivation for the definition of Killing-form?

simple-Lie-group <==> Killing-form is non-degenerate

The Killing-form of simple-Lie-group and its symmetric-space is Einstein-metric

Proof for the case of simple-Lie-group

for Lie algebra

Proof

Lie-algebra ==> δ-isometry ==> for δ-group-action ,

Because the Killing-form is the metric generated by the group action, the Lie-derivative is zero

For fields generated by

geodesic-derivative . Proof see below
curvature
. hence symmetric-space-locally
curvature
sectional-curvature for orthonormal
Ricci-curvature . hence Einstein-metric
scalar-curvature

Prop at , similarly for fields generated by (bi-invariant)

Proof

Prop

This gives

with , this gives

Proof of

need

Since group action generates , constant value ==>

need

Question any more intuitive proof?

Prop for simple-Lie-group

Integral curves of bi-invariant vector fields generated by Lie algebra are Killing-form geodesics, because

Geodesics can be written as
Assume is an integral curve of ,

Quadratic form manifold. Symmetric group of is

orbit type or
- induced metric signature (normal vector )
- isotropy-group
- quotient
- isometry of is (isometry assumed to preserve direction)
orbit type or
- induced metric signature (normal vector )
- isotropy-group
- quotient
- isometry of is

Example

spatial manifold has
spacetime quadratic manifold has and hyperboloid of one sheet

Examples of quadratic manifolds with this property

simple-Lie-group , simple-Lie-group isotropy , orbit

Lie-algebra has orthogonal decomposition , not Lie bracket decomposition

is the Lie-algebra of , is the orthogonal complement

gives the coordinates of

The Killing-form of derives the Killing-form of and the Einstein metric of

note-math