# 扭率張量

（重定向自聯絡的撓率

${\displaystyle T(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]}$

## 挠率张量

M 是切丛上带有联络 ∇ 的流形。挠率张量（有时也称为嘉当（挠率）张量）是一个向量值 2-形式，定义在向量场 XY

${\displaystyle T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]\ ,}$

### 曲率和比安基恒等式

${\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z\ .}$

${\displaystyle {\mathfrak {S}}\left(R(X,Y)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y\ .}$

1. 比安基第一恒等式

${\displaystyle {\mathfrak {S}}\left(R(X,Y)Z\right)={\mathfrak {S}}\left(T(T(X,Y),Z)+(\nabla _{X}T)(Y,Z)\right)\ ,}$

2. 比安基第二恒等式

${\displaystyle {\mathfrak {S}}\left((\nabla _{X}R)(Y,Z)+R(T(X,Y),Z)\right)=0\ .}$

### 挠率张量的分量

${\displaystyle T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}$

### 挠率形式

${\displaystyle \theta (X)=u^{-1}(d\pi (X))}$

${\displaystyle \Theta =d\theta +\omega \wedge \theta \ .}$

${\displaystyle R_{g}^{*}\Theta =g^{-1}\cdot \Theta \ ,}$

### 曲率形式与比安基恒等式

${\displaystyle \Omega =D\omega =d\omega +\omega \wedge \omega \ .}$

1. ${\displaystyle D\Theta =\Omega \wedge \theta }$
2. ${\displaystyle D\Omega =0.\,}$

${\displaystyle R(X,Y)Z=u\left(2\Omega (\pi ^{-1}(X),\pi ^{-1}(Y))\right)(u^{-1}(Z)),}$
${\displaystyle T(X,Y)=u\left(2\Theta (\pi ^{-1}(X),\pi ^{-1}(Y))\right),}$

#### 标架中的曲率形式

${\displaystyle D{\mathbf {e} }_{i}=\sum _{j=1}^{n}{\mathbf {e} }_{j}\omega _{i}^{j}\ .}$

${\displaystyle \Theta ^{k}=d\theta ^{k}+\sum _{j=1}^{n}\omega _{j}^{k}\wedge \theta ^{j}=\sum _{i,j}T_{ij}^{k}\theta ^{i}\wedge \theta ^{j}\ .}$

${\displaystyle T_{ij}^{k}=\theta ^{k}(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-[\mathbf {e} _{i},\mathbf {e} _{j}])}$

${\displaystyle {\tilde {\mathbf {e} }}_{i}=\sum _{j}\mathbf {e} _{j}g_{i}^{j}}$

${\displaystyle {\tilde {\Theta }}^{i}=(g^{-1})_{j}^{i}\Theta ^{j}.}$

${\displaystyle \Theta \in {\text{Hom}}(\wedge ^{2}TM,TM)}$

${\displaystyle \Theta =D\theta \ ,}$

### 不可约分解

${\displaystyle a_{i}=T_{ik}^{k}\ ,}$

${\displaystyle B_{jk}^{i}=T_{jk}^{i}+{\frac {1}{n-1}}\delta _{j}^{i}a_{k}-{\frac {1}{n-1}}\delta _{k}^{i}a_{j}}$

${\displaystyle T\in \operatorname {Hom} \left(\wedge ^{2}TM,TM\right)\ .}$

T 的迹 tr T，是如下定义的 T*M 中一个元素。对固定的任何向量X ∈ TMT 定义了一个 Hom(TM, TM) 中一个元素 T(X)，通过

${\displaystyle T(X):Y\mapsto T(X\wedge Y)\ .}$

${\displaystyle (\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X))\ .}$

T 不含迹的部分为

${\displaystyle T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T)}$

${\displaystyle d(\operatorname {tr} \,T)=0\ ,}$

## 特征描述与解释

### 仿射进化

${\displaystyle {\dot {C}}_{t}=\tau _{t}^{0}{\dot {x}}_{t},\quad C_{0}=0}$

${\displaystyle \tau _{t}^{0}:T_{x_{t}}M\to T_{x_{0}}M}$

### 参考标架的扭曲

${\displaystyle \left.\nabla _{\partial /\partial \tau }{\frac {\partial }{\partial x}}\right|_{x=0}=0.}$

${\displaystyle \left.T\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial \tau }}\right)\right|_{x=0}=\left.\nabla _{\frac {\partial }{\partial x}}{\frac {\partial }{\partial \tau }}\right|_{x=0}.}$

## 测地线与挠率的吸收

${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}$

• 两个联络 ∇ 与 ∇′ 具有相同的仿射参数化测地线（即相同的测地波浪），只在挠率有区别。[5]

${\displaystyle \Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}}$

${\displaystyle S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)}$
${\displaystyle A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)}$

• ${\displaystyle A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)}$  是挠率张量之差。
• ∇ 与 ∇′ 定义了相同的仿射参数化测地线族当且仅当 S(X,Y) = 0。

• 给定任何仿射联络 ∇，存在惟一一个无挠联络 ∇′ 具有共同的仿射参数化测地线。

## 注释

1. ^ See Kobayashi-Nomizu (1996) Volume 1, Proposition III.5.2.
2. ^ Kobayashi-Nomizu (1996) Volume 1, III.2.
3. ^ Kobayashi-Nomizu (1996) Volume 1, III.5.
4. ^ Goriely et al (2006).
5. ^ See Spivak (1999) Volume II, Addendum 1 to Chapter 6. See also Bishop and Goldberg (1980), section 5.10.