数量曲率

（重定向自标量曲率

${\displaystyle S={\mbox{tr}}_{g}\,\operatorname {Ric} \ .}$

${\displaystyle S=g^{ij}R_{ij}\ ,}$

${\displaystyle \operatorname {Ric} =R_{ij}\,dx^{i}\otimes dx^{j}\ .}$

${\displaystyle S=g^{ab}(\Gamma _{ab,c}^{c}-\Gamma _{ac,b}^{c}+\Gamma _{ab}^{c}\Gamma _{cd}^{d}-\Gamma _{ac}^{d}\Gamma _{bd}^{c})}$

直接几何解释

${\displaystyle {\frac {\operatorname {Vol} (B_{\varepsilon }(p)\subset M)}{\operatorname {Vol} (B_{\varepsilon }(0)\subset {\mathbb {R} }^{n})}}=1-{\frac {S}{6(n+2)}}\varepsilon ^{2}+O(\varepsilon ^{4})\ .}$

${\displaystyle {\frac {\operatorname {Area} (\partial B_{\varepsilon }(p)\subset M)}{\operatorname {Area} (\partial B_{\varepsilon }(0)\subset {\mathbb {R} }^{n})}}=1-{\frac {S}{6n}}\varepsilon ^{2}+O(\varepsilon ^{4})\ .}$

二维

${\displaystyle S={\frac {2}{\rho _{1}\rho _{2}}}\ ,}$

2 维黎曼张量只有一个独立分量，可以简单地用数量曲率和度量面积形式表示出来。在任何坐标系下，我们有

${\displaystyle 2R_{1212}\,=S\det(g_{ij})=S[g_{11}g_{22}-(g_{12})^{2}]\ .}$

传统记法

1. 黎曼曲率张量${\displaystyle R_{ijk}^{l}}$${\displaystyle R_{abcd}}$
2. 里奇张量${\displaystyle R_{ij}}$
3. 数量曲率 R