\u5047\u8bbe$(M,g)$\u662f\u9ece\u66fc\u6d41\u5f62, \u4ee4$\\tilde g=e^{2\\phi} g$, \u8fd9\u91cc$\\phi$\u662f$M$\u4e0a\u4e00\u4e2a\u5149\u6ed1\u51fd\u6570. \u8fd9\u65f6\u79f0$(M,g)$\u4e0e$(M,\\tilde g)$\u5171\u5f62.<\/p>\n
\u6211\u4eec\u611f\u5174\u8da3\u7684\u662f, \u5171\u5f62\u53d8\u6362\u4e0b\u66f2\u7387\u4e4b\u95f4\u7684\u5173\u7cfb.<\/p>\n
\u4e3a\u6b64, \u6211\u4eec\u7528\u6d3b\u52a8\u6807\u67b6\u6cd5(\u7528\u81ea\u7136\u6807\u67b6\u8ba1\u7b97\u53ef\u4ee5\u53c2\u8003\u6211\u5199\u7684Notes<\/a>). \u5047\u8bbe$\\set{e_i}$\u662f$(M,g)$\u7684\u4e00\u4e2a\u5e7a\u6b63\u6807\u67b6\u573a, $\\set{\\omega^i}$\u662f\u5176\u5bf9\u5076\u6807\u67b6\u573a. $\\nabla,\\widetilde\\nabla$\u5206\u522b\u8868\u793a\u5bf9\u5e94\u4e8e$g,\\tilde g$\u7684\u9ece\u66fc\u8054\u7edc, \u76f8\u5e94\u7684\u8054\u7edc1\u5f62\u5f0f\u8bb0\u4e3a$\\omega^i_j,\\widetilde\\omega^i_j$. (\u56de\u5fc6, \u7ed9\u5b9a\u4e00\u4e2a\u8054\u7edc$\\nabla$, \u4ee5\u53ca\u4e00\u4e2a\u5c40\u90e8\u6807\u67b6\u573a$\\set{e_i}$, \u8054\u7edc1\u5f62\u5f0f$\\set{\\omega^i_j}$\u7531\u4e0b\u5f0f\u5b9a\u4e49:$\\nabla_X(e_j)=\\omega^i_j(X)e_i.$)<\/p>\n <\/p>\n \u9ece\u66fc\u8054\u7edc\u7684\u548c\u5ea6\u91cf\u7684\u76f8\u5bb9\u6027\u5982\u4e0b:<\/p>\n $$ \u6ce8\u610f\u5230\u6211\u4eec\u59cb\u7ec8\u53ef\u9009\u53d6$\\omega^i_j=0$\u5728\u67d0\u7ed9\u5b9a\u70b9\u5904\u6210\u7acb, \u53c8\u5b9a\u4e49$\\phi_{ij}=e_j(e_i\\phi)=e_j(\\phi_i)=\\rd\\phi_i (e_j)$, \u56e0\u6b64$\\rd\\phi_i=\\phi_{ij}\\omega^j$. \u8fd9\u6837\u6839\u636e\u7ed3\u6784\u65b9\u7a0b<\/a>\u6709<\/p>\n $$\\begin{align*} \u56e0\u4e3a $\\Omega_j^i=\\frac{1}{2}R^i_{klj}\\omega^k\\wedge\\omega^l$, \u5176\u4e2d $R^i_{klj}=\\omega^i(R(e_k,e_l)e_j)$ \u4ee5\u53ca $\\widetilde \\Omega_j^i=\\frac{1}{2}\\tilde R^i_{klj}\\omega^k\\wedge\\omega^l$, \u5176\u4e2d $\\tilde R^i_{klj}=\\omega^i(\\tilde R(e_k,e_l)e_j)=\\omega^i(\\widetilde \\nabla_{e_k}\\widetilde \\nabla_{e_l}e_j-\\widetilde \\nabla_{e_l}\\widetilde \\nabla_{e_k}e_j-\\widetilde \\nabla_{[e_k,e_l]}e_j)$, $$\\begin{equation}\\label{eq:curvature13} \u7531$\\eqref{eq:curvature13}$, \u6211\u4eec\u5bb9\u6613\u5f97\u5230\u5176\u4ed6\u66f2\u7387\u7684\u5173\u7cfb:<\/p>\n $$ \u5728$\\eqref{eq:curvature04}$\u4e24\u8fb9\u540c\u65f6\u5bf9$\\tilde g^{il}$\u505a\u7f29\u5e76, \u6709<\/p>\n $$\\begin{align*} \u518d\u6b21\u7528$\\tilde g^{jk}$\u5bf9$\\eqref{eq:riccicurve}$\u505a\u7f29\u5e76, \u6709<\/p>\n $$\\begin{align*} \u6700\u540e, \u6ce8\u610f\u5230$\\phi_{,jk}$\u7684\u5b9a\u4e49, \u6211\u4eec\u6709<\/p>\n $$ \u6211\u4eec\u987a\u4fbf\u6307\u51fa, \u5728$n=2$\u7684\u60c5\u5f62, \u7531\u6b64\u7acb\u5373\u5f97\u5230\u8be5\u4e66\u7684Exercise1.72.<\/p>\n \u5047\u8bbe$(M,g)$\u662f\u9ece\u66fc\u6d41\u5f62, \u4ee4$\\tilde g=e^{2\\phi} g$, …<\/p>\n\u8054\u7edc1\u5f62\u5f0f\u7684\u5173\u7cfb<\/h4>\n
\n\\widetilde \\nabla_{e_i}\\tilde g(e_j, e_k)= \\tilde g(\\widetilde \\nabla_{e_i}e_j, e_k)+\\tilde g(e_j, \\widetilde\\nabla_{e_i}e_k)
\n$$
\n\u5373
\n$$
\n\\nabla_{e_i}(e^{2\\phi}\\delta_{jk})=e^{2\\phi}(\\widetilde \\omega^l_j(e_i)g_{lk}+\\widetilde \\omega^l_k(e_i)g_{jl}),
\n$$
\n\u4e5f\u5373
\n$$
\n2e_i(\\phi)\\delta_{jk}=2d\\phi(e_i)\\delta_{jk}=\\widetilde \\omega^l_j(e_i)\\delta_{lk}+\\widetilde \\omega^l_k(e_i)\\delta_{jl},
\n$$
\n\u6211\u4eec\u4ee4$e_i(\\phi)=\\phi_i$, \u5219
\n$$\\begin{equation}\\label{eq:tildenotorsion}
\n2\\phi_i\\delta_{jk}=\\widetilde \\omega^k_j(e_i)+\\widetilde \\omega^j_k(e_i).
\n\\end{equation}$$
\n\u7279\u522b\u5730, \u4ee4$\\phi=0$, \u5219\u5f97\u5230
\n$$\\begin{equation}\\label{eq:notorsion}
\n0=\\omega^k_j(e_i)+\\omega^j_k(e_i).
\n\\end{equation}$$
\n\u800c\u9ece\u66fc\u8054\u7edc\u7684\u65e0\u6320\u6027\u5982\u4e0b:
\n$$
\n\\nabla_{e_i}e_j-\\nabla_{e_j}e_i=[e_i,e_j]=\\widetilde \\nabla_{e_i}e_j-\\widetilde \\nabla_{e_j}e_i,
\n$$
\n\u5373
\n$$\\begin{equation}\\label{eq:2}
\n\\omega^k_j(e_i)-\\omega^k_i(e_j)=\\widetilde \\omega^k_j(e_i)-\\widetilde \\omega^k_i(e_j).
\n\\end{equation}$$
\n\u5c06$\\eqref{eq:2}$\u7684\u6307\u6807$i,j,k$\u8f6e\u6362\u5f97\u5230
\n$$\\begin{align}
\n\\omega^i_k(e_j)-\\omega^i_j(e_k)&=\\widetilde \\omega^i_k(e_j)-\\widetilde \\omega^i_j(e_k),\\label{eq:3}\\\\
\n\\omega^j_i(e_k)-\\omega^j_k(e_i)&=\\widetilde \\omega^j_i(e_k)-\\widetilde \\omega^j_k(e_i)\\label{eq:4}.
\n\\end{align}$$
\n\u5c06$\\eqref{eq:2}$+$\\eqref{eq:3}$-$\\eqref{eq:4}$, \u5e76\u5229\u7528$\\eqref{eq:notorsion}$\u5f97\u5230,
\n$$
\n-2\\omega^k_i(e_j)=\\widetilde \\omega^k_j(e_i)+\\widetilde \\omega^j_k(e_i)
\n-\\widetilde \\omega^k_i(e_j)+\\widetilde \\omega^i_k(e_j)-\\widetilde \\omega^i_j(e_k)-\\widetilde \\omega^j_i(e_k),
\n$$
\n\u518d\u5229\u7528$\\eqref{eq:tildenotorsion}$, \u5f97\u5230
\n$$
\n\\omega^k_i(e_j)=-\\phi_i\\delta_{jk}-\\phi_j\\delta_{ik}+\\phi_k\\delta_{ij}+\\widetilde\\omega^k_i(e_j),
\n$$
\n\u5373
\n$$
\n\\omega^k_i(e_j)=\\widetilde\\omega^k_i(e_j)-
\n\\phi_i\\omega^k(e_j)-\\rd\\phi(e_j)\\delta_{ik}+\\phi_k\\omega^i(e_j),
\n$$
\n\u4ece\u800c
\n$$
\n\\omega^k_i=\\widetilde\\omega^k_i-
\n\\phi_i\\omega^k-\\rd\\phi\\delta_{ik}+\\phi_k\\omega^i,
\n$$
\n\u6216\u8005, \u7b49\u4ef7\u5730
\n$$
\n\\widetilde\\omega^i_j=\\omega^i_j+
\n\\rd\\phi\\delta_{ij}-\\phi_i\\omega^j+\\phi_j\\omega^i.
\n$$<\/p>\n\u66f2\u73872\u5f62\u5f0f\u7684\u5173\u7cfb<\/h4>\n
\n\\widetilde\\Omega^i_j&=\\rd\\tilde\\omega^i_j+\\tilde\\omega^i_k\\wedge\\tilde\\omega^k_j\\\\
\n&=\\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\\\\
\n&\\qquad+(\\omega^i_k+\\rd\\phi\\delta_{ik}-\\phi_i\\omega^k+\\phi_k\\omega^i)\\wedge\\\\
\n&\\quad\\qquad(\\omega^k_j+\\rd\\phi\\delta_{kj}-\\phi_k\\omega^j+\\phi_j\\omega^k)\\\\
\n&=\\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\\\\
\n&\\qquad+\\phi_k\\phi_j\\omega^i\\wedge\\omega^k-\\sum_k\\phi_k^2\\omega^i\\wedge\\omega^j\\\\
\n&=\\Omega_j^i+\\Bigg(-\\phi_{ik}\\delta_{jl}+\\phi_{jk}\\delta_{il}+\\phi_i\\phi_k\\delta_{jl}\\\\
\n&\\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\\\\
\n&=\\Omega^i_j+(\\phi_{,jk}\\delta_{il}-\\phi_{,ik}\\delta_{jl})\\omega^k\\wedge\\omega^l.
\n\\end{align*}$$
\n\u5176\u4e2d, $\\phi_{,ij}=\\phi_{ij}-\\phi_i\\phi_j+\\frac{1}{2}\\phi_p^2\\delta_{ij}$.<\/p>\n$(1,3)$\u66f2\u7387\u5f20\u91cf\u7684\u5173\u7cfb<\/h4>\n
\n\u8fd9\u6837, \u6211\u4eec\u5f97\u5230$(1,3)$\u66f2\u7387\u5f20\u91cf\u5e94\u6ee1\u8db3\u7684\u5173\u7cfb:<\/p>\n
\n\\tilde R_{klj}^i=R^i_{klj}+(\\phi_{,jk}\\delta_{il}-\\phi_{,jl}\\delta_{ik}-\\phi_{,ik}\\delta_{jl}+\\phi_{,il}\\delta_{jk}).
\n\\end{equation}$$<\/p>\n$(0,4)$\u66f2\u7387\u5f20\u91cf\u7684\u5173\u7cfb<\/h4>\n
\n\\tilde R_{klij}=\\tilde g_{jp}\\tilde R^p_{kli}
\n=e^{2\\phi}g_{jp}\\left(
\nR^p_{kli}+(\\phi_{,ik}\\delta_{pl}-\\phi_{,il}\\delta_{pk}-\\phi_{,pk}\\delta_{il}+\\phi_{,pl}\\delta_{ik})
\n\\right),
\n$$
\n\u5373,
\n$$\\begin{equation}\\label{eq:curvature04}
\n\\tilde R_{ijkl}=e^{2\\phi}\\left(
\nR_{ijkl}+(
\n\\phi_{,ik}\\delta_{jl}-\\phi_{,il}\\delta_{jk}-\\phi_{,jk}\\delta_{il}+\\phi_{,jl}\\delta_{ik}
\n)
\n\\right).
\n\\end{equation}$$
\n\u8fd9\u4fbf\u662f$(0,4)$\u578b\u66f2\u7387\u5f20\u91cf\u7684\u5173\u7cfb.<\/p>\nRicci\u66f2\u7387\u7684\u5173\u7cfb<\/h4>\n
\n\\tilde R_{jk}&=\\tilde g^{il}\\tilde R_{ijkl}=e^{-2\\phi}g^{il}\\tilde R_{ijkl}\\\\
\n&=g^{il}\\left(
\nR_{ijkl}+(
\n\\phi_{,ik}\\delta_{jl}-\\phi_{,il}\\delta_{jk}-\\phi_{,jk}\\delta_{il}+\\phi_{,jl}\\delta_{ik}
\n)
\n\\right)\\\\
\n&=R_{jk}+\\left(
\n\\phi_{,ik}\\delta_{ji}-\\sum_i\\phi_{,ii}\\delta_{jk}-\\phi_{,jk}\\delta_{ii}+\\phi_{,ji}\\delta_{ik}\\right)\\\\
\n&=R_{jk}+\\left(
\n\\phi_{,jk}-\\sum_i\\phi_{,ii}\\delta_{jk}-n\\phi_{,jk}+\\phi_{,jk}\\right)\\\\
\n&=R_{jk}-\\left(
\n(n-2)\\phi_{,jk}+\\sum_{i}\\phi_{,ii}\\delta_{jk}
\n\\right).
\n\\end{align*}$$
\n\u4e8e\u662f\u6211\u4eec\u5f97\u5230Ricci\u66f2\u7387\u7684\u5173\u7cfb:
\n$$\\begin{equation}\\label{eq:riccicurve}
\n\\tilde R_{jk}=R_{jk}-\\left(
\n(n-2)\\phi_{,jk}+\\sum_{i}\\phi_{,ii}\\delta_{jk}
\n\\right).
\n\\end{equation}$$<\/p>\nScalar\u66f2\u7387\u7684\u5173\u7cfb<\/h4>\n
\n\\tilde R&=\\tilde g^{jk}\\tilde R_{jk}=e^{-2\\phi}g^{jk}\\tilde R_{jk}\\\\
\n&=e^{-2\\phi}g^{jk}\\left(
\nR_{jk}-\\left(
\n(n-2)\\phi_{,jk}+\\sum_{i}\\phi_{,ii}\\delta_{jk}
\n\\right)
\n\\right)\\\\
\n&=e^{-2\\phi}\\left(
\nR-2(n-1)\\sum_k\\phi_{,kk}
\n\\right),
\n\\end{align*}$$
\n\u5373
\n$$\\begin{equation}\\label{eq:scalarcurv}
\n\\tilde R=e^{-2\\phi}\\left(
\nR-2(n-1)\\sum_k\\phi_{,kk}
\n\\right).
\n\\end{equation}$$<\/p>\n\u4e0eLaplace\u7684\u5173\u7cfb<\/h4>\n
\n\\phi_{,kk}=\\phi_{kk}-\\phi_k^2+\\frac{1}{2}\\sum_p\\phi_p^2,
\n$$
\n\u56e0\u6b64,
\n$$
\n\\sum_{k}\\phi_{,kk}=\\sum_k\\phi_{kk}+\\frac{n-2}{2}\\sum_p\\phi_p^2
\n=\\Delta\\phi+\\frac{n-2}{2}|\\nabla\\phi|^2.
\n$$
\n\u8fd9\u6837, $\\eqref{eq:scalarcurv}$\u53ef\u4ee5\u5199\u6210
\n$$
\n\\tilde R=e^{-2\\phi}\\left(
\nR-2(n-1)\\Delta\\phi-(n-1)(n-2)|\\nabla\\phi|^2
\n\\right).
\n$$
\n\u8fd9\u6b63\u662fBennett Chow\u7684\u4e66Hamilton’s Ricci flow<\/em>1<\/a>\u4e2d\u7684(1.83)\u5f0f.<\/p>\n
\n\n