# 富比尼–施图迪度量

## 构造

${\displaystyle \mathbf {CP} ^{n}=\{\mathbf {Z} =[Z_{0},Z_{1},\ldots ,Z_{n}]\in {\mathbf {C} }^{n+1}\}/\{\mathbf {Z} \sim c\mathbf {Z} ,c\in \mathbf {C} ^{*}\}.}$

${\displaystyle \mathbf {C} ^{n+1}{\stackrel {(a)}{\longrightarrow }}S^{2n+1}{\stackrel {(b)}{\longrightarrow }}\mathbf {CP} ^{n}}$

### 作为度量商

'Cn+1 上标准埃尔米特度量在标准基下为

${\displaystyle ds^{2}=d\mathbf {Z} \otimes d{\overline {\mathbf {Z} }}=dZ_{0}\otimes d{\overline {Z_{0}}}+\cdots +dZ_{n}\otimes d{\overline {Z_{n}}}.\,}$

### 在局部仿射坐标中

${\displaystyle [Z_{0},\dots ,Z_{n}]\sim [1,z_{1},\dots ,z_{n}],}$

${\displaystyle h_{ij}=h(\partial _{i},\partial _{j})={\frac {(1+|\mathbf {z} |^{2})\delta _{ij}-{\bar {z}}_{i}z_{j}}{(1+|\mathbf {z} |^{2})^{2}}}.}$

${\displaystyle {\bigl (}h_{ij}{\bigr )}={\frac {1}{(1+|\mathbf {z} |^{2})^{2}}}\left[{\begin{array}{cccc}1+|\mathbf {z} |^{2}-|z_{1}|^{2}&-{\bar {z}}_{1}z_{2}&\cdots &-{\bar {z}}_{1}z_{n}\\-{\bar {z}}_{2}z_{1}&1+|\mathbf {z} |^{2}-|z_{2}|^{2}&\cdots &-{\bar {z}}_{2}z_{n}\\\vdots &\vdots &\ddots &\vdots \\-{\bar {z}}_{n}z_{1}&-{\bar {z}}_{n}z_{2}&\cdots &1+|\mathbf {z} |^{2}-|z_{n}|^{2}\end{array}}\right]}$

{\displaystyle {\begin{aligned}ds^{2}&={\frac {(1+|\mathbf {z} |^{2})|d\mathbf {z} |^{2}-({\bar {\mathbf {z} }}\cdot d\mathbf {z} )(\mathbf {z} \cdot d{\bar {\mathbf {z} }})}{(1+|\mathbf {z} |^{2})^{2}}}\\&={\frac {(1+z_{i}{\bar {z}}^{i})dz_{j}d{\bar {z}}^{j}-{\bar {z}}^{j}z_{i}dz_{j}d{\bar {z}}^{i}}{(1+z_{i}{\bar {z}}^{i})^{2}}}.\end{aligned}}}

### 在齐次坐标中

{\displaystyle {\begin{aligned}ds^{2}&={\frac {|\mathbf {Z} |^{2}|d\mathbf {Z} |^{2}-({\bar {\mathbf {Z} }}\cdot d\mathbf {Z} )(\mathbf {Z} \cdot d{\bar {\mathbf {Z} }})}{|\mathbf {Z} |^{4}}}\\&={\frac {Z_{\alpha }{\bar {Z}}^{\alpha }dZ_{\beta }d{\bar {Z}}^{\beta }-{\bar {Z}}^{\alpha }Z_{\alpha }dZ_{\beta }d{\bar {Z}}^{\beta }}{(Z_{\alpha }{\bar {Z}}^{\alpha })^{2}}}\\&=2{\frac {Z_{[\alpha }dZ_{\beta ]}{\overline {Z}}^{[\alpha }{\overline {dZ}}^{\beta ]}}{\left(Z_{\alpha }{\overline {Z}}^{\alpha }\right)^{2}}}.\end{aligned}}}

${\displaystyle Z_{[\alpha }W_{\beta ]}={\frac {1}{2}}\left(Z_{\alpha }W_{\beta }-Z_{\beta }W_{\alpha }\right).}$

${\displaystyle \omega =i\partial {\overline {\partial }}\log |\mathbf {Z} |^{2}.}$

### n = 1 情形

n = 1，有由球极投影给出的微分同胚 ${\displaystyle S^{2}\cong \mathbb {CP} ^{1}}$。这导致了特殊的霍普夫纤维化 S1S3S2。当在 CP1 中的坐标系写出富比尼–施图迪度量，它在实切丛上的限制得出 S2 上半径 1/2 的通常圆度量。

${\displaystyle ds^{2}={\frac {dz\;d{\overline {z}}}{\left(1+|z|^{2}\right)^{2}}}={\frac {dx^{2}+dy^{2}}{\left(1+r^{2}\right)^{2}}}={\frac {1}{4}}(d\phi ^{2}+\sin ^{2}\phi \,d\theta ^{2})={\frac {1}{4}}ds_{us}^{2}}$

## 曲率性质

n = 1 的特例，富比尼–施图迪度量具有恒等于 4 的数量曲率，因为它与 2-球面的圆度量等价（半径 R 球面的数量曲率是 ${\displaystyle 1/R^{2}}$）。但是，对 n > 1，富比尼–施图迪度量没有常曲率。其截面曲率由下列方程给出[1]

${\displaystyle K(\sigma )=1+3\langle JX,Y\rangle ^{2}}$

${\displaystyle Ric_{ij}=\lambda g_{ij}.}$

## 量子力学

${\displaystyle \vert \psi \rangle =\sum _{k=0}^{n}Z_{k}\vert e_{k}\rangle =[Z_{0}:Z_{1}:\ldots :Z_{n}]}$

${\displaystyle \gamma (\psi ,\phi )=\arccos {\sqrt {\frac {\langle \psi \vert \phi \rangle \;\langle \phi \vert \psi \rangle }{\langle \psi \vert \psi \rangle \;\langle \phi \vert \phi \rangle }}}}$

${\displaystyle \gamma (\psi ,\phi )=\gamma (Z,W)=\arccos {\sqrt {\frac {Z_{\alpha }{\overline {W}}^{\alpha }\;W_{\beta }{\overline {Z}}^{\beta }}{Z_{\alpha }{\overline {Z}}^{\alpha }\;W_{\beta }{\overline {W}}^{\beta }}}}.}$

${\displaystyle ds^{2}={\frac {\langle \delta \psi \vert \delta \psi \rangle }{\langle \psi \vert \psi \rangle }}-{\frac {\langle \delta \psi \vert \psi \rangle \;\langle \psi \vert \delta \psi \rangle }{{\langle \psi \vert \psi \rangle }^{2}}}.}$

## 乘积度量

${\displaystyle ds^{2}={ds_{A}}^{2}+{ds_{B}}^{2}}$

## 参考文献

1. ^ Sakai, T. Riemannian Geometry, Translations of Mathematical Monographs No. 149 (1995), American Mathematics Society.