# 问题及解答

## 第二 Bianchi 恒等式

Posted by haifeng on 2015-08-25 18:17:44 last update 2015-08-25 18:21:59 | Edit | Answers (2)

$(\nabla_X R)_{YZ}+(\nabla_Y R)_{ZX}+(\nabla_Z R)_{XY}=0.$

Ref.

1

Posted by haifeng on 2015-08-26 12:38:03

$\begin{split} (\nabla_X R)_{YZ}W &=\nabla_X (R_{YZ}W)-R_{YZ}(\nabla_X W)-R_{\nabla_X Y,Z}W-R_{Y,\nabla_X Z}W\\ &=[\nabla_X,R_{YZ}]W-R_{\nabla_X Y,Z}W-R_{Y,\nabla_X Z}W\\ \end{split}$

$\begin{split} (\nabla_Y R)_{ZX}W &=\nabla_Y (R_{ZX}W)-R_{ZX}(\nabla_Y W)-R_{\nabla_Y Z,X}W-R_{Z,\nabla_Y X}W\\ &=[\nabla_Y,R_{ZX}]W-R_{\nabla_Y Z,X}W-R_{Z,\nabla_Y X}W\\ \end{split}$

$\begin{split} (\nabla_Z R)_{XY}W &=\nabla_Z (R_{XY}W)-R_{XY}(\nabla_Z W)-R_{\nabla_Z X,Y}W-R_{X,\nabla_Z Y}W\\ &=[\nabla_Z,R_{XY}]W-R_{\nabla_Z X,Y}W-R_{X,\nabla_Z Y}W\\ \end{split}$

$\begin{split} &(\nabla_X R)_{YZ}W+(\nabla_Y R)_{ZX}W+(\nabla_Z R)_{XY}W\\ =\quad &[\nabla_X,R_{YZ}]W-R_{\nabla_X Y,Z}W-R_{Y,\nabla_X Z}W\\ +&[\nabla_Y,R_{ZX}]W-R_{\nabla_Y Z,X}W-R_{Z,\nabla_Y X}W\\ +&[\nabla_Z,R_{XY}]W-R_{\nabla_Z X,Y}W-R_{X,\nabla_Z Y}W\\ =\quad &[\nabla_X,[\nabla_Z,\nabla_Y]]W+[\nabla_X,\nabla_{[Y,Z]}]W-([\nabla_Z,\nabla_{\nabla_X Y}]+\nabla_{[\nabla_X Y,Z]})W-([\nabla_{\nabla_X Z},\nabla_Y]+\nabla_{[Y,\nabla_X Z]})W\\ +&[\nabla_Y,[\nabla_X,\nabla_Z]]W+[\nabla_Y,\nabla_{[Z,X]}]W-([\nabla_X,\nabla_{\nabla_Y Z}]+\nabla_{[\nabla_Y Z,X]})W-([\nabla_{\nabla_Y X},\nabla_Z]+\nabla_{[Z,\nabla_Y X]})W\\ +&[\nabla_Z,[\nabla_Y,\nabla_X]]W+[\nabla_Z,\nabla_{[X,Y]}]W-([\nabla_Y,\nabla_{\nabla_Z X}]+\nabla_{[\nabla_Z X,Y]})W-([\nabla_{\nabla_Z Y},\nabla_X]+\nabla_{[X,\nabla_Z Y]})W\\ \end{split}$

$\begin{split} =\quad \Bigl\{ &[\nabla_X,\nabla_{[Y,Z]}]-[\nabla_X,\nabla_{\nabla_Y Z}]-[\nabla_{\nabla_Z Y},\nabla_X]\\ + & [\nabla_Y,\nabla_{[Z,X]}]-[\nabla_Y,\nabla_{\nabla_Z X}]-[\nabla_{\nabla_X Z},\nabla_Y]\\ + & [\nabla_Z,\nabla_{[X,Y]}]-[\nabla_Z,\nabla_{\nabla_X Y}]-[\nabla_{\nabla_Y X},\nabla_Z]\Bigr\}W\\ - &\Bigl(\nabla_{[\nabla_X Y,Z]}+\nabla_{[Z,\nabla_Y X]}+\nabla_{[\nabla_Y Z,X]}+\nabla_{[X,\nabla_Z Y]}+\nabla_{[\nabla_Z X,Y]}+\nabla_{[Y,\nabla_X Z]}\Bigr)W \end{split}$

$\begin{split} &[\nabla_X,\nabla_{[Y,Z]}]-[\nabla_X,\nabla_{\nabla_Y Z}]-[\nabla_{\nabla_Z Y},\nabla_X]\\ =&[\nabla_X,\nabla_{[Y,Z]}]-([\nabla_X,\nabla_{\nabla_Y Z}]-[\nabla_X,\nabla_{\nabla_Z Y}])\\ =&[\nabla_X,\nabla_{[Y,Z]}]-[\nabla_X,\nabla_{{\nabla_Y Z}-{\nabla_Z Y}}]\\ =&[\nabla_X,\nabla_{[Y,Z]}]-[\nabla_X,\nabla_{[Y,Z]}]\\ =&0. \end{split}$

$\begin{split} =&-\Bigl(\nabla_{[\nabla_X Y,Z]}+\nabla_{[Z,\nabla_Y X]}+\nabla_{[\nabla_Y Z,X]}+\nabla_{[X,\nabla_Z Y]}+\nabla_{[\nabla_Z X,Y]}+\nabla_{[Y,\nabla_X Z]}\Bigr)W\\ =&-\Bigl(\nabla_{[\nabla_X Y-\nabla_X Y,Z]}+\nabla_{[\nabla_Y Z-\nabla_Y Z,X]}+\nabla_{[\nabla_Z X-\nabla_Z X,Y]}\Bigr)W\\ =&-\Bigl(\nabla_{[[X,Y],Z]}+\nabla_{[[Y,Z],X]}+\nabla_{[[Z,X],Y]}\Bigr)W\\ =&-\Bigl(\nabla_{[[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]}\Bigr)W\\ =&0 \end{split}$

2

Posted by haifeng on 2015-08-26 14:18:18

$\mathcal{C}(f(X,Y,Z)):=f(X,Y,Z)+f(Y,Z,X)+f(Z,X,Y).$

$\begin{split} (\nabla_Z R)(X,Y)W &=\nabla_Z(R(X,Y)W)-R(\nabla_Z X,Y)W-R(X,\nabla_Z Y)W-R(X,Y)\nabla_Z W\\ &=[\nabla_Z,R(X,Y)]W-R(\nabla_Z X,Y)W-R(X,\nabla_Z Y)W. \end{split}$

$\mathcal{C}\bigl((\nabla_X R)(Y,Z)\bigr)W=\mathcal{C}\bigl([\nabla_X,R(Y,Z)]\bigr)W-\mathcal{C}\bigl(R(\nabla_X Y,Z)\bigr)W-\mathcal{C}\bigl(R(Y,\nabla_X Z)\bigr)W.$

$\begin{split} \mathcal{C}\bigl((\nabla_X R)(Y,Z)\bigr)W&=\mathcal{C}\bigl([\nabla_X,[\nabla_Z,\nabla_Y]]\bigr)W+\mathcal{C}\bigl([\nabla_X,\nabla_{[Y,Z]}]\bigr)W-\mathcal{C}\bigl(R(\nabla_X Y,Z)\bigr)W-\mathcal{C}\bigl(R(Y,\nabla_X Z)\bigr)W\\ &=\mathcal{C}\bigl([\nabla_X,\nabla_{[Y,Z]}]\bigr)W-\mathcal{C}\bigl(R(\nabla_X Y,Z)\bigr)W+\mathcal{C}\bigl(R(\nabla_Y X,Z)\bigr)W\\ &=\mathcal{C}\bigl([\nabla_X,\nabla_{[Y,Z]}]\bigr)W-\mathcal{C}\bigl(R([X,Y],Z)\bigr)W\\ &=\mathcal{C}\bigl([\nabla_X,\nabla_{[Y,Z]}]\bigr)W-\mathcal{C}\bigl(\nabla_Z\nabla_{[X,Y]}-\nabla_{[X,Y]}\nabla_Z+\nabla_{[[X,Y],Z]}\bigr)W\\ &=\mathcal{C}\bigl([\nabla_X,\nabla_{[Y,Z]}]\bigr)W-\mathcal{C}\bigl([\nabla_Z,\nabla_{[X,Y]}]\bigr)W-\mathcal{C}\bigl(\nabla_{[[X,Y],Z]}\bigr)W\\ &=\mathcal{C}\bigl([\nabla_X,\nabla_{[Y,Z]}]\bigr)W-\mathcal{C}\bigl([\nabla_Z,\nabla_{[X,Y]}]\bigr)W\\ &=0. \end{split}$

$(\nabla_X R)(Y,Z)+(\nabla_Y R)(Z,X)+(\nabla_Z R)(X,Y)=0.$