cf. wiki:Symmetric_space wiki:Simple_Lie_group
[symmetric-space-locally] :=
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 ๅค็ๅ็ฉบ้ด็
็ถๅๅฎไน ๅค็ metric ้่ฟ action ็ๆ็ๅ ถๅฎๅฐๆน็ metric, ่ไธๆฏ bi-invariant ็ i.e. ็พคไฝ็จ็ไธค็งๅฝขๅผ้ฝ็ปๅบ็ธๅ็ metric
่ฟๆ ท็ๅฎไนไฝฟๅพ็พคไฝ็จๆฏ Killing-form ็ isometry
ไธๆฏ simple-Lie-group ไนๅฏไปฅๅฎไน Killing-form
Question Killing-form ็ๅฎไน็ๅจๆบ?
simple-Lie-group <==> Killing-form ้้ๅ
simple-Lie-group and its symmetric-space ็ Killing-form ๆฏ Einstein-metric
Proof of simple-Lie-group ็ๆ ๅต
- for Lie algebra
Proof
Lie-algebra ==> ฮด-isometry ==> for ฮด-group-action ,
ๅ ไธบ Killing-form ๆฏ็พคไฝ็จ็ๆ็ metric ๆไปฅ Lie-derivative ๆฏ้ถ
ๅฏนไบ ็ๆ็ๅบ
-
geodesic-derivative . Proof see below
-
curvature
-
. hence symmetric-space-locally
-
curvature
-
sectional-curvature for orthonormal
-
Ricci-curvature . hence Einstein-metric
-
scalar-curvature
Prop at , ๅ็ๅฏน ็ๆ็ๅบ (bi-invariant)
Proof
Prop
่ฟ็ปๅบ
with , ่ฟ็ปๅบ
Proof of
need
็ฑไบ็พคไฝ็จ็ๆ , ๅธธๅผ ==>
need
need
by
Question any more intuitive proof?
Prop for simple-Lie-group
Lie algebra ็ๆ็ bi-invariant vector field ็็งฏๅๆฒ็บฟ้ฝๆฏ Killing-form ๆตๅฐ็บฟ, ๅ ไธบ
- ๆตๅฐ็บฟๅฏไปฅๅไธบ
- ๅ่ฎพ ๆฏ ็็งฏๅๆฒ็บฟ
ไบๆฌกๅๆตๅฝข. ็ๅฏน็งฐ็พค
-
orbit type or
- induced metric signature (normal vector )
- isotropy-group
- quotient
- isometry of is (isometry ๅ่ฎพไฟๆๆนๅ)
-
orbit type or
- induced metric signature (normal vector )
- isotropy-group
- quotient
- isometry of is
Example
-
spatial manifold ๆ
-
ๆถ็ฉบไบๆฌกๅๆตๅฝขๆ ๅๅๅถๅๆฒ้ข
ไบๆฌกๅๆตๅฝข็ไพๅญๆ่ฟ็งๆง่ดจ
simple-Lie-group , simple-Lie-group isotropy , orbit
Lie-algebra ๆๆญฃไบคๅ่งฃ , ไธๆฏ Lie bracket ๅ่งฃ
ๆฏ ็ Lie-algebra, ๆฏๆญฃไบค่กฅ
็ปๅบ ็ๅๆ
็ Killing-form ๅฏผๅบ ็ Killing-form ๅ ็ Einstein metric