$\\newcommand{\\rd}{\\mathrm{d}}\\newcommand{\\set}[1]{\\left\\{#1\\right\\}}\\newcommand{\\Lp}{\\Delta\\,}$\u4e8b\u5b9e\u4e0a, \u5047\u8bbe$\\rd s^2=g_{ij}\\rd x^i\\rd x^j$\u662f$M^2=(\\Omega,g)$\u4e0a\u7684Riemann\u5ea6\u91cf. \u8981\u4f7f$M^2$ \u662f\u5171\u5f62\u5e73\u5766\u7684, \u90a3\u4e48
\n \\[
\n \\rd s^2=g^{ij}\\rd x_i\\rd x_j=e^{2\\lambda u}\\left((\\rd x_1)^2+(\\rd x_2)^2\\right).
\n \\]
\n \u4e0b\u9762, \u6211\u4eec\u7528\u6d3b\u52a8\u6807\u67b6\u6cd5\u6765\u8ba1\u7b97$M^2$\u7684\u9ad8\u65af\u66f2\u7387$K$.
\n \u4ee4
\n \\[
\n \\rd\\omega_i=e^{\\lambda u}\\rd x_i,\\quad i=1,2,
\n \\]
\n \u5219, $\\set{\\omega_i}_{i=1,2}$\u662f\u4e00\u4e2a\u5e7a\u6b63\u4f59\u6807\u67b6\u573a, \u4e14
\n \\begin{align*}
\n \\rd\\omega_1&=-\\lambda e^{\\lambda u}u_2\\rd x_1\\wedge\\rd x_2=-\\lambda u_2\\rd x_1\\wedge\\omega_2\\\\
\n \\rd\\omega_2&=\\lambda e^{\\lambda u}u_1\\rd x_1\\wedge\\rd x_2=-\\lambda u_1\\rd x_2\\wedge\\omega_1,
\n \\end{align*}
\n \u7531\u5e7a\u6b63\u6807\u67b6\u4e0b\u7684Cartan\u7ed3\u6784\u65b9\u7a0b,
\n \\[\\begin{cases}
\n \\rd\\omega_i=-\\omega_{ij}\\wedge\\omega_j\\\\
\n \\rd\\omega_{ij}=-\\omega_{ik}\\wedge\\omega_{kj}+\\Omega_{ij},
\n \\end{cases}\\]
\n \u5176\u4e2d, $\\Omega_{ij}=\\frac{1}{2}R_{ijkl}\\omega_k\\wedge\\omega_l$, $R_{ijkl}=-R_{ijlk}$, \u6211\u4eec\u6709
\n \\[
\n \\omega_{12}=\\lambda\\left(u_2\\rd x_1-u_1\\rd x_2\\right)=-\\omega_{21}.
\n \\]
\n \u56e0$\\omega_{12}=-\\omega_{21}$, \u4e8e\u662f\u7531Cartan\u57fa\u672c\u5f15\u7406\u5f97\u5230$\\omega_{12}$\u662f\u552f\u4e00\u7684. \u518d\u6b21\u5229\u7528Cartan\u7ed3\u6784\u65b9\u7a0b, \u6211\u4eec\u6709
\n \\[
\n \\rd\\omega_{12}=\\frac{1}{2}R_{12kl}\\omega_k\\wedge\\omega_l=-\\lambda\\Lp u\\rd x_1\\wedge\\rd x_2=-\\lambda\\frac{\\Lp u}{e^{2\\lambda u}}\\omega_1\\wedge\\omega_2,
\n \\]
\n \u4ece\u800c
\n \\[
\n K=R_{1212}=-\\lambda\\frac{\\Lp u}{e^{2\\lambda u}},
\n \\]
\n \u8fd9\u5373\u662f\u8bf4
\n \\[
\n \\Lp u+e^{2\\lambda u}K\/\\lambda=0.
\n \\]
\n \u6700\u540e, \u4ee4$\\lambda=1\/2$, \u6211\u4eec\u6709
\n \\[
\n \\Lp u+2K e^{u}=0.
\n \\] <\/p>\n","protected":false},"excerpt":{"rendered":"
$\\newcommand{\\rd}{\\mathrm{d}}\\newcommand …<\/p>\n