note-math

sectional-curvature_(tag)

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

截面曲率是二次型 (可能退化) 限制在 $⋀^{2} 𝑉^{⊺}$ 方向空间 i.e. 限制在单位长度

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

curvature 可以恢复自 sectional-curvature. Proof 不需要非退化, 对称双线性可以恢复自二次型 quadratic-form

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. 曲率只有纯量部分且纯量曲率是常值

Proof

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

<==> $𝑅 - 𝜆 \cdot \frac{1}{2} \cdot 𝑔 ⧀ 𝑔$ 是零二次型

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

$curvature$ 正交分解给出 $𝑅 = \frac{1}{2 𝑛 (𝑛 - 1)} \cdot scal \cdot 𝑔 ⧀ 𝑔$ 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 三维 $𝑊 = 0$ + (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) 二次型流形 $ℚ^{𝑝, 𝑞} (\pm 𝑎^{2})$ 有 constant-sectional-curvature $𝑅 = \frac{1}{\pm 2 𝑎^{2}} 𝑔 ⧀ 𝑔$

Proof

使用子流形技术. 子流形 $ℚ (𝑝, 𝑞) (\pm 𝑎^{2})$ 上的点 $𝑥$ 在 $ℝ^{𝑝, 𝑞}$ 有切空间与法空间

$𝑥 \in ℚ (𝑝, 𝑞) (\pm 𝑎^{2})$ 点的子流形测地线坐标 + 法空间作为流形 $ℝ^{𝑝, 𝑞}$ 坐标

$ℝ^{𝑝, 𝑞}$ 在此坐标在 $𝑥$ 点的 coordinate-frame $\partial_{𝑖}$ orthonormal

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

曲率的 metric-dual $𝑔 [\nabla_{𝑖}, \nabla_{𝑖^{'}}] = 𝑔 [\partial_{𝑖} + Γ_{𝑖}, \partial_{𝑖^{'}} + Γ_{𝑖^{'}}]$

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

$ℝ^{𝑝, 𝑞}$ 的曲率是零 $𝑅_{𝑖 𝑗 𝑖^{'} 𝑗^{'}} = 0$

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

二次型流形 co-dimension 1, 法空间 dimension 1, 法向场 $Γ_{𝑖 𝑗}^{⟂} = 𝜆 𝑛$ with 单位法向场 $𝑛$

\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}

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

在 $ℝ^{𝑝, 𝑞}$ 普通坐标在 $𝑥$ 点 $𝑛 = \frac{grad | 𝑥 |^{2}}{| grad | 𝑥 |^{2} |} = \frac{1}{𝑎} 𝑥^{𝑖} \partial_{𝑖}$ 且 $\nabla_{\partial_{𝑖}} 𝑛 = \partial_{𝑖} 𝑛 = \frac{1}{𝑎} \partial_{𝑖}$

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

==>

\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}

宇宙常数 $Λ = \pm \frac{(𝑛 - 1) (𝑛 - 2)}{2 𝑎^{2}}$

二次型流形中的 Lorentz 流形 $ℚ^{1, 𝑛} (- 𝑎^{2}), ℚ^{2, 𝑛 - 1} (𝑎^{2})$ 有 "静态坐标", i.e. 在静态坐标中 metric 将是静态形式

$ℚ^{1, 𝑛} (- 𝑎^{2})$ 静态坐标 :=

分解到半径 $𝑟$ + 双曲线 $ℚ^{1, 1} (- 𝑎^{2} + 𝑟^{2})$ + 球面 $ℚ^{0, 𝑛 - 1} (- 𝑟^{2})$

坐标 $(𝑡, 𝑟, 𝕊^{𝑛 - 2})$ 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 将是

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}})

$ℚ^{2, 𝑛 - 1} (𝑎^{2})$ 静态坐标 :=

分解到半径 $𝑟$ + 球面 $ℚ^{2, 0} (𝑎^{2} + 𝑟^{2})$ + 球面 $ℚ^{0, 𝑛 - 2} (- 𝑟^{2})$

坐标 $(𝑡, 𝑟, 𝕊^{𝑛 - 2})$ 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 将是

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}})

$ℚ^{2, 𝑛 - 1} (𝑎^{2})$ 的时间轴的行为存在 $𝕊^{1}$ like. 而且存在 closed time-like geodesicm, 从而不 causal

"单叶双曲面" $ℚ^{1, 𝑛} (- 𝑎^{2})$ 的时间轴的行为是 $ℝ$ like, 空间存在 $𝕊^{𝑛 - 1}$ like. 存在 closed space-like geodesic

$ℚ^{1, 𝑛} (- 𝑎^{2})$ 可以 "时间切片" 化为 $ℝ \times 𝕊^{𝑛 - 1}$ . $sinh$ 是 $ℝ$ 的微分同胚

\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}}

$ℚ^{2, 𝑛 - 1} (𝑎^{2})$ "可视化" 的例子: $ℝ^{3}$ 中的 $ℚ^{2, 1} (1)$ or $𝑥_{1}^{2} + 𝑥_{2}^{2} - 𝑥_{3}^{2} = 1$ , 单叶双曲面

虽然 $ℚ^{2, 1} (1)$ 类时测地线总是闭合的, 表现为椭圆, 但类时非测地线可以无限长度, 例如可以不断逼近类光测地线

类光测地线表现为 "抛物线" …