note-math

symmetric-space-locally_(tag) := $\nabla 𝑅=0$

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 := $𝑔\left(𝑋,𝑌\right)≔\pm \text{ tr }\left[𝑋,\left[𝑌,\right]\right]=\pm \text{tr}\left(\text{ad }𝑋\text{ ad }𝑌\right)$ for $𝑋,𝑌$ in the tangent space at $\mathrm{𝟙}$

Then define the metric at $\mathrm{𝟙}$ 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

Prop ${\nabla }_{𝑋}𝑌=\frac{1}{2}\left[𝑋,𝑌\right]$ at $\mathrm{𝟙}$, similarly for fields generated by $𝑋,𝑌$ (bi-invariant)

Proof

Prop for simple-Lie-group

Quadratic form manifold. Symmetric group of ${\mathrm{ℝ}}^{𝑝,𝑞}$ is $\text{SO}\left(𝑝,𝑞\right)$

• orbit type $|𝑥{|}^2=1$ or ${\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(1\right)$

  • induced metric signature $\left(𝑝-1,𝑞\right)$ (normal vector $|𝑥{|}^2=1>0$)
  • isotropy-group $\text{SO}\left(𝑝-1,𝑞\right)$
  • quotient $\frac{\text{SO}\left(𝑝,𝑞\right)}{\text{SO}\left(𝑝-1,𝑞\right)}={\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(1\right)$
  • isometry of ${\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(1\right)$ is $\text{SO}\left(𝑝,𝑞\right)$ (isometry assumed to preserve direction)

• orbit type $|𝑥{|}^2=-1$ or ${\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(-1\right)$

  • induced metric signature $\left(𝑝,𝑞-1\right)$ (normal vector $|𝑥{|}^2=-1<0$)
  • isotropy-group $\text{SO}\left(𝑝,𝑞-1\right)$
  • quotient $\frac{\text{SO}\left(𝑝,𝑞\right)}{\text{SO}\left(𝑝,𝑞-1\right)}={\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(-1\right)$
  • isometry of ${\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(-1\right)$ is $\text{SO}\left(𝑝,𝑞\right)$

${\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(\pm 1\right)={\mathrm{\mathbb{Q}}}^{𝑞,𝑝}\left(\mp 1\right)$

Example

• $\left(0,𝑛\right)$ spatial manifold has ${\mathrm{\mathbb{Q}}}^{1,𝑛}\left(1\right)={\mathrm{ℍ}\mathrm{𝕪}}^{𝑛}=\frac{\text{SO}\left(1,𝑛\right)}{\text{SO}\left(𝑛\right)},{\mathrm{\mathbb{Q}}}^{0,𝑛+1}\left(-1\right)={\mathrm{𝕊}}^{𝑛}=\frac{\text{SO}\left(𝑛+1\right)}{\text{SO}\left(𝑛\right)}$

• $\left(1,𝑛-1\right)$ spacetime quadratic manifold has ${\mathrm{\mathbb{Q}}}^{2,𝑛-1}\left(1\right)=\frac{\text{SO}\left(2,𝑛-1\right)}{\text{SO}\left(1,𝑛-1\right)}$ and hyperboloid of one sheet ${\mathrm{\mathbb{Q}}}^{1,𝑛}\left(-1\right)=\frac{\text{SO}\left(1,𝑛\right)}{\text{SO}\left(1,𝑛-1\right)}$

Examples of quadratic manifolds with this property

simple-Lie-group $𝐺$, simple-Lie-group isotropy $𝐻$, orbit $\frac{𝐺}{𝐻}$

Lie-algebra has orthogonal decomposition $\text{g }=\text{ h }\oplus \frac{\text{ g}}{\text{ h}}$, not Lie bracket decomposition

$\text{h}$ is the Lie-algebra of $𝐻$, $\frac{\text{g }}{\text{h }}={\text{ h}}^{⟂}$ is the orthogonal complement

$\text{exp}$ gives the coordinates of $𝐺,𝐻,\frac{𝐺}{𝐻}$

The Killing-form of $\text{g}$ derives the Killing-form of $\text{h}$ and the Einstein metric of $\frac{\text{g}}{\text{h}}$

• $\left[\text{h},\text{h}\right]\subset \text{ h}$

• $\left[\text{h},\frac{\text{g}}{\text{h}}\right]\subset \frac{\text{ g}}{\text{ h}}$

• $\left[\frac{\text{g}}{\text{h}},\frac{\text{g}}{\text{h}}\right]\subset \text{ h}$