note-math

sectional-curvature_(tag)

According to #link(<symmetry-of-curvature>)[], $curvature \subset ⨀^{2} ⋀^{2} 𝑉^{⊺}$

Sectional curvature is a quadratic form (possibly degenerate) restricted to the direction space $⋀^{2} 𝑉^{⊺}$ i.e. restricted to unit length

𝐾 (𝑋 \land 𝑌) = \frac{𝑅 (𝑋 \land 𝑌, 𝑋 \land 𝑌)}{| 𝑋 \land 𝑌 |^{2}}

Curvature can be recovered from sectional-curvature. Proof does not require non-degeneracy, symmetric bilinear forms can be recovered from quadratic forms

Prop $\frac{1}{2} \cdot (𝑔 ⧀ 𝑔) (𝑋 \land 𝑌, 𝑋 \land 𝑌) = | 𝑋 \land 𝑌 |^{2}$

constant-sectional-curvature_(tag) $𝐾 (𝑋 \land 𝑌) = const$ <==>

(𝑅 = \frac{1}{2 𝑛 (𝑛 - 1)} \cdot scal \cdot 𝑔 ⧀ 𝑔) \land (scal = const)

i.e. the curvature only has a scalar part and the scalar curvature is constant

Proof

constant-sectional-curvature <==> $𝑅 (𝑋 \land 𝑌, 𝑋 \land 𝑌) = 𝜆 | 𝑋 \land 𝑌 |^{2} = 𝜆 \cdot \frac{1}{2} \cdot (𝑔 ⧀ 𝑔) (𝑋 \land 𝑌, 𝑋 \land 𝑌)$

<==> $𝑅 - 𝜆 \cdot \frac{1}{2} \cdot 𝑔 ⧀ 𝑔$ is a zero quadratic form

<==> $𝑅 = 𝜆 \cdot \frac{1}{2} \cdot 𝑔 ⧀ 𝑔$

The orthogonal decomposition of $curvature$ gives $𝑅 = \frac{1}{2 𝑛 (𝑛 - 1)} \cdot scal \cdot 𝑔 ⧀ 𝑔$ with $scal = constant$

constant-sectional-curvature-imply-Einstein-metric_(tag)

Proof trace-free Ricci-curvature = 0

constant-sectional-curvature-low-dimension_(tag)

$dim = 2$ ==> constant-sectional-curvature = Einstein-metric = constant-scalar-curvature
$dim = 3$ ==> constant-sectional-curvature = Einstein-metric Proof 3D $𝑊 = 0$ + (Einstein <==> $tr-free-Ric = 0$ )

quadratic-manifold_(tag) :=

ℚ^{𝑝, 𝑞} (\pm 1) = {𝑥 \in ℝ^{𝑝, 𝑞} : {⟨ 𝑥 ⟩}^{2} = \pm 1}

where ${⟨ 𝑥 ⟩}^{2} = \sum_{sign} sign \sum_{𝑖_{sign}} 𝑥_{𝑖_{sign}}^{2} = 𝑥_{1}^{2} + \dots + 𝑥_{𝑝}^{2} - (𝑥_{𝑝 + 1}^{2} + \dots + 𝑥_{𝑝 + 𝑞}^{2})$

quadratic-manifold-is-constant-sectional-curvature_(tag) Quadratic manifold $ℚ^{𝑝, 𝑞} (\pm 𝑎^{2})$ has constant-sectional-curvature $𝑅 = \frac{1}{\pm 2 𝑎^{2}} 𝑔 ⧀ 𝑔$

Proof

Using submanifold techniques. A point $𝑥$ on the submanifold $ℚ (𝑝, 𝑞) (\pm 𝑎^{2})$ has tangent space and normal space in $ℝ^{𝑝, 𝑞}$

Submanifold geodesic coordinates for the point $𝑥 \in ℚ (𝑝, 𝑞) (\pm 𝑎^{2})$ + normal space as coordinates for the manifold $ℝ^{𝑝, 𝑞}$

The coordinate-frame $\partial_{𝑖}$ of $ℝ^{𝑝, 𝑞}$ at point $𝑥$ in these coordinates is orthonormal

Separate tangent, normal, $Γ = Γ^{⊤} + Γ^{⟂}$

The metric-dual of curvature $𝑔 [\nabla_{𝑖}, \nabla_{𝑖^{'}}] = 𝑔 [\partial_{𝑖} + Γ_{𝑖}, \partial_{𝑖^{'}} + Γ_{𝑖^{'}}]$

==> $𝑅_{𝑖 𝑗 𝑖^{'} 𝑗^{'}}^{⊤} = 𝑅_{𝑖 𝑗 𝑖^{'} 𝑗^{'}} + 𝑔 (Γ_{𝑖 𝑗}^{⟂}, Γ_{𝑖^{'} 𝑗^{'}}^{⟂}) - 𝑔 (Γ_{𝑖 𝑗^{'}}^{⟂}, Γ_{𝑖^{'} 𝑗}^{⟂})$

The curvature of $ℝ^{𝑝, 𝑞}$ is zero $𝑅_{𝑖 𝑗 𝑖^{'} 𝑗^{'}} = 0$

==> $𝑅_{𝑖 𝑗 𝑖^{'} 𝑗^{'}}^{⊤} = 𝑔 (Γ_{𝑖 𝑗}^{⟂}, Γ_{𝑖^{'} 𝑗^{'}}^{⟂}) - 𝑔 (Γ_{𝑖 𝑗^{'}}^{⟂}, Γ_{𝑖^{'} 𝑗}^{⟂})$

Quadratic manifold co-dimension 1, normal space dimension 1, normal field $Γ_{𝑖 𝑗}^{⟂} = 𝜆 𝑛$ with unit normal field $𝑛$

\begin{matrix} 𝑔 (Γ_{𝑖 𝑗}^{⟂}, 𝑛) & = & 𝑔 (Γ_{𝑖 𝑗}, 𝑛) & by Γ_{𝑖 𝑗}^{⊤} ⟂ 𝑛 ⟹ 𝑔 (Γ_{𝑖 𝑗}, 𝑛) = 0 \\ = & 𝑔 (\nabla_{\partial_{𝑖}} \partial_{𝑗}, 𝑛) \\ = & \partial_{𝑖} (𝑔 (\partial_{𝑗}, 𝑛)) - 𝑔 (\partial_{𝑗}, \nabla_{\partial_{𝑖}} 𝑛) \\ = & - 𝑔 (\partial_{𝑗}, \nabla_{\partial_{𝑖}} 𝑛) & by \partial_{𝑗} ⟂ 𝑛 ⟹ 𝑔 (𝑛, \partial_{𝑗}) = 0 \end{matrix}

So $Γ_{𝑖 𝑗}^{⟂} = - 𝑔 (\nabla_{\partial_{𝑖}} 𝑛, \partial_{𝑗}) 𝑛$

In $ℝ^{𝑝, 𝑞}$ ordinary coordinates at point $𝑥$ , $𝑛 = \frac{grad | 𝑥 |^{2}}{| grad | 𝑥 |^{2} |} = \frac{1}{𝑎} 𝑥^{𝑖} \partial_{𝑖}$ and $\nabla_{\partial_{𝑖}} 𝑛 = \partial_{𝑖} 𝑛 = \frac{1}{𝑎} \partial_{𝑖}$

==> $Γ_{𝑖 𝑗}^{⟂} = - 𝑔 (\nabla_{\partial_{𝑖}} 𝑛, \partial_{𝑗}) 𝑛 = - \frac{1}{𝑎} 𝑔 (\partial_{𝑖}, \partial_{𝑗}) 𝑛$

==>

\begin{matrix} 𝑅_{𝑖 𝑗 𝑖^{'} 𝑗^{'}}^{⊤} & = & \frac{1}{𝑎^{2}} 𝑔 (𝑛, 𝑛) (𝑔_{𝑖 𝑗} 𝑔_{𝑖^{'} 𝑗^{'}} - 𝑔_{𝑖 𝑗^{'}} 𝑔_{𝑖^{'} 𝑗}) \\ = & \frac{1}{\pm 2 𝑎^{2}} {(𝑔 ⧀ 𝑔)}_{𝑖 𝑗 𝑖^{'} 𝑗^{'}} \end{matrix}

Cosmological constant $Λ = \pm \frac{(𝑛 - 1) (𝑛 - 2)}{2 𝑎^{2}}$

Lorentz manifolds $ℚ^{1, 𝑛} (- 𝑎^{2}), ℚ^{2, 𝑛 - 1} (𝑎^{2})$ in quadratic manifolds have "static coordinates", i.e. the metric will be in static form in static coordinates

$ℚ^{1, 𝑛} (- 𝑎^{2})$ static coordinates :=

Decomposition into radius $𝑟$ + hyperbola $ℚ^{1, 1} (- 𝑎^{2} + 𝑟^{2})$ + sphere $ℚ^{0, 𝑛 - 1} (- 𝑟^{2})$

Coordinates $(𝑡, 𝑟, 𝕊^{𝑛 - 2})$ with

\begin{matrix} 𝑟^{2} & = & 𝑥_{3}^{2} + \dots + 𝑥_{𝑛 + 1}^{2} \\ 𝑥_{1} & = & {(𝑎^{2} - 𝑟^{2})}^{\frac{1}{2}} sinh (\frac{1}{𝑎} 𝑡) \\ 𝑥_{2} & = & {(𝑎^{2} - 𝑟^{2})}^{\frac{1}{2}} cosh (\frac{1}{𝑎} 𝑡) \\ 𝑥_{𝑖} & = & 𝑟 \frac{𝑥_{𝑖}}{𝑟} \end{matrix}

metric will be

𝑔 = (1 - \frac{𝑟^{2}}{𝑎^{2}}) 𝑑 𝑡^{2} - ({(1 - \frac{𝑟^{2}}{𝑎^{2}})}^{- 1} 𝑑 𝑟^{2} + 𝑟^{2} 𝑔_{𝕊^{𝑛 - 2}})

$ℚ^{2, 𝑛 - 1} (𝑎^{2})$ static coordinates :=

Decomposition into radius $𝑟$ + sphere $ℚ^{2, 0} (𝑎^{2} + 𝑟^{2})$ + sphere $ℚ^{0, 𝑛 - 2} (- 𝑟^{2})$

Coordinates $(𝑡, 𝑟, 𝕊^{𝑛 - 2})$ with

\begin{matrix} 𝑟^{2} & = & 𝑥_{3}^{2} + \dots + 𝑥_{𝑛 + 1}^{2} \\ 𝑥_{1} & = & {(𝑎^{2} + 𝑟^{2})}^{\frac{1}{2}} cos (\frac{1}{𝑎} 𝑡) \\ 𝑥_{2} & = & {(𝑎^{2} + 𝑟^{2})}^{\frac{1}{2}} sin (\frac{1}{𝑎} 𝑡) \\ 𝑥_{𝑖} & = & 𝑟 \frac{𝑥_{𝑖}}{𝑟} \end{matrix}

metric will be

𝑔 = (1 + \frac{𝑟^{2}}{𝑎^{2}}) 𝑑 𝑡^{2} - ({(1 + \frac{𝑟^{2}}{𝑎^{2}})}^{- 1} 𝑑 𝑟^{2} + 𝑟^{2} 𝑔_{𝕊^{𝑛 - 2}})

The behavior of the time axis of $ℚ^{2, 𝑛 - 1} (𝑎^{2})$ is $𝕊^{1}$ like. And there exists closed time-like geodesicm, hence not causal

The time axis behavior of the "single-sheeted hyperboloid" $ℚ^{1, 𝑛} (- 𝑎^{2})$ is $ℝ$ like, and the space existence is $𝕊^{𝑛 - 1}$ like. There exists closed space-like geodesic

$ℚ^{1, 𝑛} (- 𝑎^{2})$ can be "time-sliced" into $ℝ \times 𝕊^{𝑛 - 1}$ . $sinh$ is a diffeomorphism of $ℝ$

\begin{matrix} 𝑥_{1} & = & 𝑎 sinh (\frac{1}{𝑎} 𝑡) \\ 𝑥_{𝑖} & = & 𝑎 cosh (\frac{1}{𝑎} 𝑡) \frac{𝑥_{𝑖}}{𝑎 cosh (\frac{1}{𝑎} 𝑡)} \end{matrix}

metric

𝑔 = 𝑑 𝑡^{2} - 𝑎^{2} {cosh}^{2} (\frac{1}{𝑎} 𝑡) 𝑔_{𝕊^{𝑛}}

Example of "visualization" of $ℚ^{2, 𝑛 - 1} (𝑎^{2})$ : $ℚ^{2, 1} (1)$ in $ℝ^{3}$ or $𝑥_{1}^{2} + 𝑥_{2}^{2} - 𝑥_{3}^{2} = 1$ , single-sheeted hyperboloid

Although time-like geodesics of $ℚ^{2, 1} (1)$ are always closed, appearing as ellipses, time-like non-geodesics can have infinite length, for example, they can continuously approach light-like geodesics

Light-like geodesics appear as "parabolas" …