# 黎曼曲率張量

${\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w.}$

${\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w}$

## 對稱性和恆等式

• ${\displaystyle R:(w,u,v)\rightarrow R(u,v)w,}$

• ${\displaystyle R=R_{kij}^{l}dx^{k}\otimes {\frac {\partial }{\partial x^{l}}}\otimes dx^{i}\otimes dx^{j}.}$

• ${\displaystyle R:(w,z,u,v)\rightarrow g(R(u,v)w,z),}$

• ${\displaystyle R=R_{klij}dx^{k}\otimes dx^{l}\otimes dx^{i}\otimes dx^{j}.}$
• 注：上述纺射联络空间 ${\displaystyle (M,\nabla )}$上的曲率张量 ${\displaystyle R}$与黎曼流形 ${\displaystyle (M,g)}$ 上的黎曼曲率张量 ${\displaystyle R}$ 是同一个对象的不同表现形式.
• ${\displaystyle R_{klij}=g_{lm}R_{kij}^{m}}$.

• ${\displaystyle R(u,v)=-R(v,u)_{}^{}}$
• ${\displaystyle \langle R(u,v)w,z\rangle =-\langle R(u,v)z,w\rangle _{}^{}}$
• ${\displaystyle R(u,v)w+R(v,w)u+R(w,u)v=0_{}^{}}$

${\displaystyle \langle R(u,v)w,z\rangle =\langle R(w,z)u,v\rangle _{}^{}}$

${\displaystyle \nabla _{u}R(v,w)+\nabla _{v}R(w,u)+\nabla _{w}R(u,v)=0}$

• ${\displaystyle R_{abcd}=-R_{bacd}\,}$
• ${\displaystyle R_{abcd}=R_{cdab}\,}$
• 第一（代數）比安基恒等式：${\displaystyle R_{abcd}+R_{adbc}+R_{acdb}=0\,}$或等價地寫為${\displaystyle R_{a[bcd]}=0\,}$
• 第二（微分）比安基恒等式：${\displaystyle \nabla _{e}R_{abcd}+\nabla _{d}R_{abec}+\nabla _{c}R_{abde}=0\,}$或等價地寫為${\displaystyle R_{ab[cd;e]}=0\,}$