共形变换下曲率关系的活动标架计算方法

联络1形式的关系

$$\widetilde \nabla_{e_i}\tilde g(e_j, e_k)= \tilde g(\widetilde \nabla_{e_i}e_j, e_k)+\tilde g(e_j, \widetilde\nabla_{e_i}e_k)$$

$$\nabla_{e_i}(e^{2\phi}\delta_{jk})=e^{2\phi}(\widetilde \omega^l_j(e_i)g_{lk}+\widetilde \omega^l_k(e_i)g_{jl}),$$

$$2e_i(\phi)\delta_{jk}=2d\phi(e_i)\delta_{jk}=\widetilde \omega^l_j(e_i)\delta_{lk}+\widetilde \omega^l_k(e_i)\delta_{jl},$$

$$\label{eq:tildenotorsion} 2\phi_i\delta_{jk}=\widetilde \omega^k_j(e_i)+\widetilde \omega^j_k(e_i).$$

$$\label{eq:notorsion} 0=\omega^k_j(e_i)+\omega^j_k(e_i).$$

$$\nabla_{e_i}e_j-\nabla_{e_j}e_i=[e_i,e_j]=\widetilde \nabla_{e_i}e_j-\widetilde \nabla_{e_j}e_i,$$

$$\label{eq:2} \omega^k_j(e_i)-\omega^k_i(e_j)=\widetilde \omega^k_j(e_i)-\widetilde \omega^k_i(e_j).$$

\begin{align} \omega^i_k(e_j)-\omega^i_j(e_k)&=\widetilde \omega^i_k(e_j)-\widetilde \omega^i_j(e_k),\label{eq:3}\\ \omega^j_i(e_k)-\omega^j_k(e_i)&=\widetilde \omega^j_i(e_k)-\widetilde \omega^j_k(e_i)\label{eq:4}. \end{align}

$$-2\omega^k_i(e_j)=\widetilde \omega^k_j(e_i)+\widetilde \omega^j_k(e_i) -\widetilde \omega^k_i(e_j)+\widetilde \omega^i_k(e_j)-\widetilde \omega^i_j(e_k)-\widetilde \omega^j_i(e_k),$$

$$\omega^k_i(e_j)=-\phi_i\delta_{jk}-\phi_j\delta_{ik}+\phi_k\delta_{ij}+\widetilde\omega^k_i(e_j),$$

$$\omega^k_i(e_j)=\widetilde\omega^k_i(e_j)- \phi_i\omega^k(e_j)-\rd\phi(e_j)\delta_{ik}+\phi_k\omega^i(e_j),$$

$$\omega^k_i=\widetilde\omega^k_i- \phi_i\omega^k-\rd\phi\delta_{ik}+\phi_k\omega^i,$$

$$\widetilde\omega^i_j=\omega^i_j+ \rd\phi\delta_{ij}-\phi_i\omega^j+\phi_j\omega^i.$$

曲率2形式的关系

\begin{align*} \widetilde\Omega^i_j&=\rd\tilde\omega^i_j+\tilde\omega^i_k\wedge\tilde\omega^k_j\\ &=\rd\omega^i_j-\rd\phi_i\wedge\omega^j-\phi_i\rd\omega^j+\rd\phi_j\wedge\omega^i+\phi_j\rd\omega^i\\ &\qquad+(\omega^i_k+\rd\phi\delta_{ik}-\phi_i\omega^k+\phi_k\omega^i)\wedge\\ &\quad\qquad(\omega^k_j+\rd\phi\delta_{kj}-\phi_k\omega^j+\phi_j\omega^k)\\ &=\Omega^i_j-\phi_{ik}\omega^k\wedge\omega^j+\phi_{jk}\omega^k\wedge\omega^i+\phi_i\phi_k\omega^k\wedge\omega^j\\ &\qquad+\phi_k\phi_j\omega^i\wedge\omega^k-\sum_k\phi_k^2\omega^i\wedge\omega^j\\ &=\Omega_j^i+\Bigg(-\phi_{ik}\delta_{jl}+\phi_{jk}\delta_{il}+\phi_i\phi_k\delta_{jl}\\ &\qquad\qquad\qquad\qquad-\phi_{k}\phi_j\delta_{il}-\sum_p\phi_p^2\delta_{ik}\delta_{jl}\Bigg)\omega^k\wedge\omega^l\\ &=\Omega^i_j+(\phi_{,jk}\delta_{il}-\phi_{,ik}\delta_{jl})\omega^k\wedge\omega^l. \end{align*}

$(1,3)$曲率张量的关系

$$\label{eq:curvature13} \tilde R_{klj}^i=R^i_{klj}+(\phi_{,jk}\delta_{il}-\phi_{,jl}\delta_{ik}-\phi_{,ik}\delta_{jl}+\phi_{,il}\delta_{jk}).$$

$(0,4)$曲率张量的关系

$$\tilde R_{klij}=\tilde g_{jp}\tilde R^p_{kli} =e^{2\phi}g_{jp}\left( R^p_{kli}+(\phi_{,ik}\delta_{pl}-\phi_{,il}\delta_{pk}-\phi_{,pk}\delta_{il}+\phi_{,pl}\delta_{ik}) \right),$$

$$\label{eq:curvature04} \tilde R_{ijkl}=e^{2\phi}\left( R_{ijkl}+( \phi_{,ik}\delta_{jl}-\phi_{,il}\delta_{jk}-\phi_{,jk}\delta_{il}+\phi_{,jl}\delta_{ik} ) \right).$$

Ricci曲率的关系

\begin{align*} \tilde R_{jk}&=\tilde g^{il}\tilde R_{ijkl}=e^{-2\phi}g^{il}\tilde R_{ijkl}\\ &=g^{il}\left( R_{ijkl}+( \phi_{,ik}\delta_{jl}-\phi_{,il}\delta_{jk}-\phi_{,jk}\delta_{il}+\phi_{,jl}\delta_{ik} ) \right)\\ &=R_{jk}+\left( \phi_{,ik}\delta_{ji}-\sum_i\phi_{,ii}\delta_{jk}-\phi_{,jk}\delta_{ii}+\phi_{,ji}\delta_{ik}\right)\\ &=R_{jk}+\left( \phi_{,jk}-\sum_i\phi_{,ii}\delta_{jk}-n\phi_{,jk}+\phi_{,jk}\right)\\ &=R_{jk}-\left( (n-2)\phi_{,jk}+\sum_{i}\phi_{,ii}\delta_{jk} \right). \end{align*}

$$\label{eq:riccicurve} \tilde R_{jk}=R_{jk}-\left( (n-2)\phi_{,jk}+\sum_{i}\phi_{,ii}\delta_{jk} \right).$$

Scalar曲率的关系

\begin{align*} \tilde R&=\tilde g^{jk}\tilde R_{jk}=e^{-2\phi}g^{jk}\tilde R_{jk}\\ &=e^{-2\phi}g^{jk}\left( R_{jk}-\left( (n-2)\phi_{,jk}+\sum_{i}\phi_{,ii}\delta_{jk} \right) \right)\\ &=e^{-2\phi}\left( R-2(n-1)\sum_k\phi_{,kk} \right), \end{align*}

$$\label{eq:scalarcurv} \tilde R=e^{-2\phi}\left( R-2(n-1)\sum_k\phi_{,kk} \right).$$

与Laplace的关系

$$\phi_{,kk}=\phi_{kk}-\phi_k^2+\frac{1}{2}\sum_p\phi_p^2,$$

$$\sum_{k}\phi_{,kk}=\sum_k\phi_{kk}+\frac{n-2}{2}\sum_p\phi_p^2 =\Delta\phi+\frac{n-2}{2}|\nabla\phi|^2.$$

$$\tilde R=e^{-2\phi}\left( R-2(n-1)\Delta\phi-(n-1)(n-2)|\nabla\phi|^2 \right).$$

1. Chow, Bennett, Peng Lu, and Lei Ni. Hamilton’s Ricci flow. Vol. 77. American Mathematical Soc., 2006.

1. lttt(小小泪)

计算曲率大致有两个办法: 一个是在自然基下来计算; 一个是用活动标架. 关于活动标架的计算办法, 可以参考沈一兵老师的< 黎曼几何初步>.

2. 尚潇

你好，请问你学习《微分几何》用的是什么教材？我遇到一个棘手的问题，在计算黎曼曲率的时候，借用坐标基底场，这个我已经求解了，现在我想采用非坐标基底场求解，书上说采用标架计算曲率，大致是说嘉当（Certan）第一结构方程和第二结构方程，可是这个我着实不太懂，真心想请教你一下，可否留个邮箱或者QQ,谢谢~~~