{"id":769,"date":"2021-07-14T01:08:30","date_gmt":"2021-07-14T01:08:30","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=769"},"modified":"2021-07-15T06:35:09","modified_gmt":"2021-07-15T06:35:09","slug":"guanyulimanjihedeyixiesikaoti","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=769","title":{"rendered":"\u5173\u4e8e\u9ece\u66fc\u51e0\u4f55\u7684\u4e00\u4e9b\u601d\u8003\u9898"},"content":{"rendered":"


\n

\u6458\u8981<\/span>.<\/span> \u8fd9\u662f\u5b66\u4e60\u9ece\u66fc\u51e0\u4f55\u7684\u4e00\u4e9b\u601d\u8003\u9898, \u4e3b\u8981\u53c2\u8003\u4e66\u76ee\u4e3a\u4f0d\u9e3f\u7199\u7b49\u7f16\u8457\u7684\u300a\u9ece\u66fc\u51e0\u4f55\u521d\u6b65\u300b. \u9898\u76ee\u6765\u6e90\u4e8e\u674e\u5149\u6c49\u5728\u5929\u5143\u897f\u5357\u6570\u5b66\u4e2d\u5fc32021\u9752\u5e74\u6559\u5e08\u6691\u671f\u5b66\u6821\u7684\u8bb2\u4e49.
\n<\/div>
\n
\n1. \u5fae\u5206\u6d41\u5f62\u3001\u8054\u7edc\u4e0e\u66f2\u7387<\/a><\/span>\n\n
  1. \u5047\u8bbe$(M^n,g)$\u662f\u53ef\u5b9a\u5411\u9ece\u66fc\u6d41\u5f62, $X\\in\\mathfrak{X}(M)$, $(U;x^i)$\u662f\u5b9a\u5411\u76f8\u7b26\u7684\u5c40\u90e8\u5750\u6807\u7cfb, \u4ee4
    \n\\[
    \n\\omega=\\sum_{i=1}^n(-1)^{i+1}\\sqrt{g}X^idx^1\\wedge\\cdots\\wedge \\widehat{dx^I}\\wedge\\cdots\\wedge dx^n,
    \n\\]
    \n\u5176\u4e2d$g=\\det(g_{ij})$, $X=X^i\\frac{\\partial}{\\partial x^i}$. \u8bc1\u660e:
    1. $\\omega$\u662f$M$\u4e0a\u6574\u4f53\u5b9a\u4e49\u7684$(n-1)$\u6b21\u5916\u5fae\u5206\u5f62\u5f0f;<\/li>
    2. $i_XdV_M=\\omega$, \u5176\u4e2d$i_X$\u4e3a\u5185\u4e58, \u5b9a\u4e49\u4e3a\u5bf9\u4efb\u610f\u7684$n$\u6b21\u5916\u5fae\u5206\u5f0f$\\phi$, \u4ee5\u53ca\u4efb\u610f\u7684$X_1,\\ldots,x_{n-1}\\in\\mathfrak{X}(M)$, \u6709
      \n\\[
      \ni_X\\phi(X_1,\\ldots,X_{n-1})=\\phi(X,X_1,\\ldots,X_{n-1}).
      \n\\]<\/li><\/ol><\/li>
    3. \u5047\u8bbe$M$\u662f\u5d4c\u5165\u5230$\\mathbb{R}^{n+1}$\u4e2d\u7684\u8d85\u66f2\u9762, $\\{X^A\\}$\u662f$\\mathbb{R}^{n+1}$\u4e2d\u7684\u76f4\u89d2\u5750\u6807\u7cfb, \u5bf9\u4efb\u610f\u7684$P\\in M$, \u5b58\u5728$P$\u7684\u5f00\u9886\u57df$U$, \u4f7f\u5f97$M\\cap U$\u6709\u8868\u793a$X^A=f^A(u^1,\\ldots, u^n)$, $1\\leq A\\leq n+1$, $(u^1,\\ldots,u^n)\\in D\\subset\\mathbb{R}^{n}$, \u5176\u4e2d$D$\u4e3a$\\mathbb{R}^n$\u4e2d\u7684\u5f00\u96c6.
      1. \u8bc1\u660e: $M$\u4e0a\u5355\u4f4d\u6cd5\u5411\u91cf$\\xi=\\xi^A\\frac{\\partial}{\\partial x^A}$\u7684\u5206\u91cf$\\xi^A=\\frac{w^A}{w}$, \u5176\u4e2d
        \n\\[
        \nw^A=(-1)^{A+1}\\frac{\\partial(f^1,\\ldots,\\hat{f^A},\\ldots,f^{n+1})}{\\partial(u^1,\\ldots,u^n)},\\quad
        \nw=\\left(\\sum_{A=1}^{n+1}(w^A)^2\\right)^{1\/2};
        \n\\]<\/li>
      2. \u6c42$\\mathbb{R}^{n+1}$\u5728$M$\u4e0a\u7684\u8bf1\u5bfc\u5ea6\u91cf$g=g_{ij}du^i\\otimes du^j$, \u4e14\u8bc1\u660e
        \n\\[
        \ng=\\det(g_{ij})=w^2.
        \n\\]<\/li>
      3. \u8bc1\u660e: $dV_M=i_\\xi(dx^1\\wedge\\cdots\\wedge dx^{n+1})|_M$.<\/li><\/ol><\/li>
      4. \u5047\u8bbe$(M_1,g_1)$, $(M_2,g_2)$\u662f\u4e24\u4e2a\u9ece\u66fc\u6d41\u5f62, \u4ee4$M=M_1\\times M_2$, \u5bf9\u4efb\u610f\u7684$(x,y)\\in M$, \u8bbe$\\pi_i:M\\to M_i$, $i=1,2$, \u4e3a\u81ea\u7136\u6295\u5f71, \u5b9a\u4e49\u6620\u5c04$\\alpha_i: M_i\\to M$, \u4f7f\u5f97\u5bf9\u4efb\u610f\u7684$z\\in M_1$, \u6709$\\alpha_1(z)=(z,y)$; \u4ee5\u53ca\u5bf9\u4efb\u610f\u7684$z\\in M_2$, \u6709$\\alpha_2(z)=(x,z)$. \u663e\u7136\u6709$\\pi_i\\circ\\alpha_i=id_{M_i}: M_i\\to M_i$, \u4ee5\u53ca$M_(x,y)=(\\alpha_1)_{*,x}M_{1,x}\\oplus(\\alpha_2)_{*,y}M_{2,y}=M_{1,x}\\oplus M_{2,y}$.
        1. \u8bc1\u660e$M$\u4e0a\u5177\u6709\u4e58\u79ef\u5ea6\u91cf$g=g_1\\times g_2$, \u5176\u5b9a\u4e49\u4e3a
          \n\\[
          \ng\\left((\\alpha_1)_*X_1+(\\alpha_2)_*Y_1,(\\alpha_1)_*X_2+(\\alpha_2)_*Y_2\\right)=g_1(X_1,X_2)+g_2(Y_1,Y_2),
          \n\\]
          \n\u5176\u4e2d$X_1,X_2\\in M_{1,x}$, $Y_1,Y_2\\in M_{2,y}$;<\/li>
        2. \u82e5$(U;x^i)$, $(V;y^\\alpha)$\u5206\u522b\u662f$M_1$\u4e0e$M_2$\u7684\u5c40\u90e8\u5750\u6807\u7cfb, ${}^1D$, ${}^2D$\u5206\u522b\u662f$M_1$, $M_2$\u7684\u9ece\u66fc\u8054\u7edc, \u5bf9\u5e94\u7684Christoffel\u7b26\u53f7\u5206\u522b\u8bb0\u4e3a${}^1\\Gamma_{ij}^k$, ${}^2\\Gamma_{\\alpha\\beta}^\\gamma$. $M$\u4e0a\u7684\u9ece\u66fc\u8054\u7edc\u4e0eChristoffel\u7b26\u53f7\u5206\u522b\u8bb0\u4e3a$D$\u4e0e$\\Gamma_{AB}^C$, \u5219
          \n\\[
          \n\\begin{cases}
          \nD_{\\frac{\\partial}{\\partial x^i}}\\frac{\\partial}{\\partial x^j}=\\left({}^1\\Gamma_{ji}^k\\circ \\pi_1\\right)\\frac{\\partial}{\\partial x^k},\\\\
          \nD_{\\frac{\\partial}{\\partial y^\\alpha}}\\frac{\\partial}{\\partial y^\\beta}=\\left({}^1\\Gamma_{\\beta\\alpha}^\\gamma\\circ \\pi_2\\right)\\frac{\\partial}{\\partial y^\\gamma},
          \n\\end{cases}\\quad
          \n\\begin{cases}
          \nD_{\\frac{\\partial}{\\partial y^\\alpha}}\\frac{\\partial}{\\partial x^i}=0,\\\\
          \nD_{\\frac{\\partial}{\\partial x^i}}\\frac{\\partial}{\\partial y^\\alpha}=0.
          \n\\end{cases}\\]<\/li><\/ol><\/li>
        3. \u5047\u8bbe$M$\u662f$n$\u4e3a\u5149\u6ed1\u6d41\u5f62, $g$\u4e0e$\\tilde g$\u90fd\u662f$M$\u4e0a\u7684\u5149\u6ed1\u9ece\u66fc\u5ea6\u91cf, $\\lambda\\in C^\\infty(M)$, \u4e14$\\tilde g=\\lambda^2g$, $\\lambda>0$. $\\tilde g$\u79f0\u4e3a$g$\u7684\u5171\u5f62\u5ea6\u91cf. \u5047\u8bbe$(U;x^i)$\u662f$M$\u7684\u5c40\u90e8\u5750\u6807\u7cfb.
          1. \u82e5$\\Gamma_{ij}^k$\u4e0e$\\tilde{\\Gamma}_{ij}^k$\u5206\u522b\u662f$g$\u4e0e$\\tilde{g}$\u7684Christoffel\u7b26\u53f7, \u5219
            \n\\[
            \n\\tilde{\\Gamma}_{ij}^k=\\Gamma_{ij}^k+\\delta_{i}^k\\partial_j(\\ln\\lambda)+\\delta_{j}^k\\partial_i(\\ln\\lambda)-g_{ij}g^{kl}\\partial_l(\\ln\\lambda).
            \n\\]
            \n\u7279\u522b\u5730, \u5f53$\\lambda=e^\\rho$, $\\rho\\in C^\\infty(M)$\u65f6,
            \n\\[
            \n\\tilde{\\Gamma}_{ij}^k=\\Gamma_{ij}^k+\\delta_{i}^k\\partial_j\\rho+\\delta_{j}^k\\partial_i\\rho-g_{ij}g^{kl}\\partial_l\\rho.
            \n\\]<\/li>
          2. \u5047\u8bbe$\\Delta_g$\u4e0e$\\Delta_{\\tilde{g}}$\u5206\u522b\u662f$g$\u4e0e$\\tilde{g}$\u7684Beltrami–Laplace\u7b97\u5b50, \u8bc1\u660e:
            \n\\[
            \n\\Delta_{\\tilde{g}}f=\\lambda^{-2}\\left( \\Delta_{g}f+n-2g(\\nabla(\\ln\\lambda),\\nabla f) \\right),
            \n\\]
            \n\u5176\u4e2d$f\\in C^\\infty(M)$\u4e3a\u4efb\u610f\u5149\u6ed1\u51fd\u6570.<\/li><\/ol><\/li>
          3. \u5728\u4e09\u7ef4\u7403\u9762$S^3$\u4e0a\u5b9a\u4e49\u5e7a\u6b63\u6807\u67b6$\\vec i,\\vec j,\\vec j$, \u4f7f\u5f97
            \n\\[
            \n[\\vec i,\\vec j]=\\vec k,\\quad[\\vec j,\\vec k]=\\vec i,\\quad [\\vec k,\\vec i]=\\vec j.
            \n\\]
            \n\u6c42$S^3$\u5728\u8be5\u5e7a\u6b63\u6807\u67b6\u573a\u4e0b\u7684\u8054\u7edc\u7cfb\u6570.<\/li>
          4. \u5047\u8bbe$(M,g)$\u4e3a\u9ece\u66fc\u6d41\u5f62, \u82e5\u5bf9\u4efb\u610f\u7684$x,y\\in M$, $M$\u4e2d\u4ece$x$\u5230$y$\u7684\u5e73\u884c\u79fb\u52a8\u4e0e\u8fde\u63a5$x$\u5230$y$\u7684\u66f2\u7ebf\u6bb5\u65e0\u5173, \u5219$M$\u7684\u66f2\u7387\u5f20\u91cf\u6052\u4e3a\u96f6.<\/li>
          5. \u5047\u8bbe$(M,g)$\u662f$n(\\geq3)$\u7ef4\u8fde\u901a\u9ece\u66fc\u6d41\u5f62, \u82e5\u5bf9\u4efb\u610f\u7684$X,Y,Z,W\\in \\mathfrak{X}$\u6ee1\u8db3\u6052\u7b49\u5f0f:
            \n\\[
            \nR(X,Y,Z,W)=\\frac{1}{n}\\left\\{ \\mathrm{Ric}(X,W)g(Y,Z)-\\mathrm{Ric}(X,Z)g(Y,W) \\right\\},
            \n\\]
            \n\u8bc1\u660e$(M,g)$\u662f\u5e38\u66f2\u7387\u7a7a\u95f4.<\/li>
          6. \u5047\u8bbe$(M,g)$\u662f$n(\\geq3)$\u7ef4\u8fde\u901a\u9ece\u66fc\u6d41\u5f62, \u4e14\u6709$\\lambda\\in C^\\infty(M)$, \u4f7f\u5f97$\\mathrm{Ric}=\\lambda g$, \u8bc1\u660e
            1. $M$\u662fEinstein\u6d41\u5f62, \u5373$M$\u7684\u6570\u91cf\u66f2\u7387\u4e3a\u5e38\u6570;<\/li>
            2. \u5f53$n=3$\u65f6, $M$\u662f\u5e38\u66f2\u7387\u7a7a\u95f4;<\/li>
            3. \u82e5$M$\u7684\u6570\u91cf\u66f2\u7387$S\\neq0$, \u5219$M$\u4e0a\u4e0d\u5b58\u5728\u975e\u96f6\u5e73\u884c\u5411\u91cf\u573a.<\/li><\/ol><\/li>
            4. \u5047\u8bbe$(M,g)$\u662f\u9ece\u66fc\u6d41\u5f62, \u5728$M_x\\times M_x$\u4e0a\u5b9a\u4e49
              \n\\[
              \nQ(X\\wedge Y,Z\\wedge W)=R(X,Y,Z,W).
              \n\\]
              1. \u8bc1\u660e$Q$\u662f\u5bf9\u79f0\u7684;<\/li>
              2. \u82e5$Q$\u5728\u6bcf\u4e00\u70b9$x\\in M$\u6b63\u5b9a(\u8d1f\u5b9a), \u5219$M$\u6709\u8d1f(\u6b63)\u7684\u622a\u9762\u66f2\u7387;<\/li>
              3. \u82e5$M$\u6709\u8d1f(\u6b63)\u7684\u622a\u9762\u66f2\u7387, $Q$\u662f\u5426\u6b63(\u8d1f\u5b9a).<\/li><\/ol><\/li>
              4. \u5982\u679c$(M_1,g_1)$\u4e0e$(M_2,g_2)$\u5747\u662f\u5e38\u66f2\u7387\u7a7a\u95f4, \u5219$M_1\\times M_2$\u5728\u5ea6\u91cf$g=g_1+g_2$\u4e0b\u662f\u5e38\u66f2\u7387\u7a7a\u95f4\u5417?<\/li><\/ol>\n
                \n2. \u5b8c\u5907\u6027\u3001\u5c40\u90e8\u5bf9\u79f0\u7a7a\u95f4\u3001\u5bf9\u79f0\u7a7a\u95f4\u3001\u5b50\u6d41\u5f62\u3001\u6d4b\u5730\u7ebf\u4e0eJacobi\u573a<\/a><\/span>\n\n
                1. \u5047\u8bbe$(M,g)$\u4e3a\u9ece\u66fc\u6d41\u5f62, \u5149\u6ed1\u66f2\u7ebf$\\gamma:[0,+\\infty)\\to M$\u79f0\u4e3a\u53d1\u6563\u66f2\u7ebf, \u5982\u679c\u5bf9\u4e8e\u4efb\u610f\u7d27\u81f4\u5b50\u96c6$K\\subset M$, \u5fc5\u5b58\u5728$t_0>0$, \u4f7f\u5f97$\\gamma(t_0)\\not\\in K$, \u5176\u957f\u5ea6\u5b9a\u4e49\u4e3a
                  \n \\[
                  \n \\lim_{s\\to\\infty}\\int_0^s\\lvert \\gamma'(t) \\rvert dt.
                  \n \\]
                  \n \u8bc1\u660e: $M$\u662f\u5b8c\u5907\u7684\u5f53\u4e14\u4ec5\u5f53$M$\u4e0a\u7684\u6bcf\u4e00\u6761\u53d1\u6563\u66f2\u7ebf\u90fd\u6709\u65e0\u9650\u957f\u5ea6.
                  \n <\/li>
                2. \u5047\u8bbe$(M,g)$\u4e3a\u9ece\u66fc\u6d41\u5f62, \u4ece$P\\in M$\u51fa\u53d1\u7684\u5f27\u957f\u53c2\u6570(\u6b63\u89c4)\u6d4b\u5730\u7ebf$\\gamma:[0,+\\infty)\\to M$\u79f0\u4e3a\u4ece$P$\u51fa\u53d1\u7684\u5c04\u7ebf, \u5982\u679c$\\gamma(0)=P$, \u4e14\u5bf9\u4efb\u610f\u7684$t\\in(0,+\\infty)$, $\\gamma|_{[0,t]}$\u662f\u8fde\u63a5$P$\u4e0e$\\gamma(t)$\u7684\u6700\u77ed\u66f2\u7ebf, \u5373$d(P,\\gamma(t))=t$. \u8bc1\u660e: \u5982\u679c$M$\u662f\u5b8c\u5907\u975e\u7d27\u7684, \u5219\u5bf9\u4efb\u610f\u7684$P\\in M$, \u5728$M$\u4e0a\u5fc5\u5b58\u5728\u4ece$P$\u51fa\u53d1\u7684\u5c04\u7ebf.
                  \n <\/li>
                3. \u5047\u8bbe$(M,g)$\u662f\u9ece\u66fc\u6d41\u5f62, \u4e14$DR\\equiv0$, \u8fd9\u662f\u6d41\u5f62\u79f0\u4e3a\u662f\u5c40\u90e8\u5bf9\u79f0\u7a7a\u95f4.
                  \n
                  1. \u5047\u8bbe$\\gamma:[0,b]\\to M$\u662f\u6d4b\u5730\u7ebf, $X,Y,Z$\u662f\u6cbf\u7740$\\gamma$\u7684\u5e73\u884c\u5411\u91cf\u573a, \u5219$R(X,Y)Z$\u4e5f\u6cbf\u7740$\\gamma$\u5e73\u884c;
                    \n <\/li>
                  2. \u5982\u679c$M$\u8fde\u901a\u4e14$\\dim M=2$, \u5219$M$\u662f\u5e38\u66f2\u7387\u7a7a\u95f4.
                    \n <\/li><\/ol>
                  3. \u5047\u8bbe$M$\u662f\u9ece\u66fc\u6d41\u5f62, $a: M\\to M$\u662f\u7b49\u8ddd, \u4e14\u5b58\u5728$P\\in M$, \u4f7f\u5f97$\\sigma(P)=P$, $\\sigma_{*P}=-\\mathrm{id}|_{M_P}$. \u53c8\u8bbe$\\gamma:(-\\epsilon,\\epsilon)\\to M$\u662f\u8fc7$P$\u70b9\u7684\u6d4b\u5730\u7ebf, \u5373$\\gamma(0)=P$, $X(t)$\u662f\u6cbf\u7740$\\gamma(t)$\u7684\u5e73\u884c\u5411\u91cf\u573a, \u5219
                    \n \\[
                    \n \\sigma_{*\\gamma(t)}(X|_{\\gamma(t)})=-X|_{\\gamma(-t)},\\quad \\forall t\\in(-\\epsilon,\\epsilon).
                    \n \\]
                    \n <\/li>
                  4. \u5047\u8bbe$M=\\mathbb{R}_+^2=\\left\\{ (x,y)\\in \\mathbb{R}^2: y>0 \\right\\}$. \u53d6\u5ea6\u91cf$g$\u4f7f\u5f97
                    \n \\[
                    \n g_{11}=g_{22}=\\frac{1}{y^2},\\quad g_{12}=0.
                    \n \\]
                    \n
                    1. \u6c42$\\mathbb{R}_+^2$\u7684\u5ea6\u91cf$g$\u5728$(x,y)$\u4e0b\u7684Christoffel\u8bb0\u53f7;
                      \n <\/li>
                    2. \u6c42$\\mathbb{R}_+^2$\u7684\u6d4b\u5730\u7ebf;
                      \n <\/li>
                    3. \u8bc1\u660e$(M,g)$\u662f\u5b8c\u5907\u7684;
                      \n <\/li>
                    4. \u6c42$(M,g)$\u7684\u622a\u9762\u66f2\u7387.
                      \n <\/li><\/ol>
                    5. \u5047\u8bbe$(M, D)$\u662f\u4eff\u5c04\u8054\u7edc\u7a7a\u95f4, $\\left\\{ e_i \\right\\}$\u4e0e$\\left\\{ \\omega^i \\right\\}$\u5206\u522b\u662f\u4e92\u4e3a\u5bf9\u5076\u7684\u5c40\u90e8\u6807\u67b6\u573a. \u5bf9\u4efb\u610f\u7684$\\theta\\in A^r(M)$, \u82e5$D$\u662f\u65e0\u6320\u8054\u7edc, \u5219
                      \n
                      1. $d\\theta=\\sum_{i=1}^n\\omega^i\\wedge D_{e_i}\\theta$;
                        \n <\/li>
                      2. $d\\theta(X_1,\\ldots,X_{r+1})=\\sum_{r=1}^{n+1}(-1)^{i+1}(D_{X_i}\\theta)(X_1,\\ldots,\\hat{X_i},\\ldots,X_{r+1})$, \u5176\u4e2d$X_1,\\ldots,X_{r+1}\\in \\mathfrak{X}(M)$.
                        \n <\/li><\/ol>
                      3. \u5982\u679c\u5bf9\u4efb\u610f\u7684$x\\in M$, \u90fd\u5b58\u5728\u7b49\u8ddd$\\phi: M\\to M$, \u5b83\u662f\u5173\u4e8e$x$\u70b9\u7684\u53cd\u5c04, \u5373\u5bf9\u4efb\u610f\u7684\u8fc7$x$\u7684\u6d4b\u5730\u7ebf$\\gamma$, $\\gamma(0)=x$, \u5728$\\gamma(t)$\u7684\u5b9a\u4e49\u57df\u5185, \u603b\u6709$\\gamma(\\gamma(t))=\\gamma(-t)$. \u6b64\u65f6\u79f0$M$\u4e3a\u5bf9\u79f0\u7a7a\u95f4. \u8bc1\u660e:
                        \n
                        1. $M$\u662f\u5b8c\u5907\u7684;
                          \n <\/li>
                        2. $M$\u662f\u9f50\u6027\u7684, \u5373\u5bf9\u4efb\u610f\u7684$x,y\\in M$, \u5b58\u5728\u7b49\u8ddd$\\varphi: M\\to M$, \u4f7f\u5f97$\\varphi(x)=y$;
                          \n <\/li>
                        3. $M$\u662f\u5c40\u90e8\u5bf9\u79f0\u7a7a\u95f4.
                          \n <\/li><\/ol>
                        4. \u5047\u8bbe$M$\u662f$\\bar{M}$\u7684\u5b50\u6d41\u5f62, \u8bc1\u660e: $M$\u5728$\\bar{M}$\u4e2d\u662f\u5168\u6d4b\u5730\u5b50\u6d41\u5f62, \u5f53\u4e14\u4ec5\u5f53\u5207\u4e1b$TM$\u5173\u4e8e$\\bar{M}$\u7684Levi-Civita\u8054\u7edc$\\bar{D}$\u5e73\u884c, \u5373\u4efb\u610f\u66f2\u7ebf$\\gamma:[0,1]\\to M$, \u7528$P^\\gamma$\u8868\u793a$\\bar{D}$\u7684\u5e73\u79fb\u540c\u6784, \u6709$P^\\gamma(M_{\\gamma(0)}=M_{\\gamma(1)}$.
                          \n <\/li>
                        5. \u5047\u8bbe$\\gamma:[0,+\\infty)\\to M$\u662f\u5c40\u90e8\u5bf9\u79f0\u7a7a\u95f4$(M,g)$\u7684\u6d4b\u5730\u7ebf, $P=\\gamma(0)$, \u5b9a\u4e49\u7ebf\u6027\u6620\u5c04$K_t: M_{\\gamma(t)}\\to M_{\\gamma(t)}$\u4e3a
                          \n \\[
                          \n K_t(W)=R(W,\\gamma'(t))\\gamma'(t),\\quad \\forall W\\in M_{\\gamma(t)}.
                          \n \\]
                          \n \u8bc1\u660e:
                          \n
                          1. $K_t$\u662f\u81ea\u5171\u8f6d\u6620\u5c04, \u5373$\\langle K_t(v_1),v_2 \\rangle=\\langle v_1,K_t(v_2) \\rangle$, \u5176\u4e2d$v_1,v_2\\in M_{\\gamma(t)}$;
                            \n <\/li>
                          2. \u53d6$M_{\\gamma(0)}$\u7684\u5e7a\u6b63\u57fa$\\left\\{ e_i \\right\\}$\u4f7f\u5f97$K_0$\u5bf9\u89d2\u5316, \u5373$K_0(e_i)=\\lambda_i e_i$, \u5e76\u628a$\\left\\{ e_i \\right\\}$\u6cbf\u7740$\\gamma(t)$\u5e73\u884c\u79fb\u52a8\u5f97\u5230$\\left\\{ e_i(t) \\right\\}$, \u5219\u6709
                            \n\t\\[
                            \n\t K_t(e_i(t))=\\lambda_ie_i(t),\\quad \\forall t\\in[0,+\\infty).
                            \n\t\\]
                            \n <\/li>
                          3. \u5047\u8bbe$J(t)=J^i(t)e_i(t)$, \u5219$J(t)$\u662f\u6cbf\u7740$\\gamma(t)$\u7684Jacobi\u573a\u5f53\u4e14\u4ec5\u5f53
                            \n\t\\[
                            \n\t \\frac{d^2J^i(t)}{dt^2}+\\lambda_iJ^i(t)=0,\\quad i=1,\\ldots, n;
                            \n\t\\]
                            \n <\/li>
                          4. \u70b9$P=\\gamma(0)$\u6cbf\u7740$\\gamma(t)$\u7684\u5171\u8f6d\u70b9\u662f$\\gamma\\left( k\\pi\/\\sqrt{\\lambda} \\right)$, \u5176\u4e2d$k$\u662f\u6b63\u6574\u6570, $\\lambda$\u662f$K_0$\u7684\u6b63\u7279\u5f81\u503c.
                            \n <\/li><\/ol>
                          5. \u5047\u8bbe$M$\u662f\u5b8c\u5907\u7684\u4e8c\u7ef4\u9ece\u66fc\u6d41\u5f62, $\\gamma:[0,+\\infty)\\to M$\u662f\u6d4b\u5730\u7ebf. $K(t)$\u662f$M$\u6cbf\u7740$\\gamma(t)$\u7684Gauss\u66f2\u7387, $L(t)$\u662f\u5b9a\u4e49\u5728$[0,+\\infty)$\u4e0a\u7684\u5149\u6ed1\u51fd\u6570, $t_0\\in(0,+\\infty)$. \u53c8\u8bbe
                            \n
                            1. \u5b58\u5728\u6b63\u5e38Jacobi\u573a$J(t)$\u6ee1\u8db3$J(0)=J(t_0)=0$, \u4f46\u5728$(0,t_0)$\u5185\u90e8, $J(t)\\neq0$;
                              \n <\/li>
                            2. \u5bf9\u4efb\u610f\u7684$t\\in[0,+\\infty)$, $K(t)\\leq L(t)$.
                              \n <\/li><\/ol> \u8bc1\u660e\u65b9\u7a0b
                              \n \\[
                              \n \\tilde{f}”(t)+L(t)\\tilde{f}(t)=0
                              \n \\]
                              \n \u7684\u6bcf\u4e2a\u89e3$\\tilde{f}(t)$\u5728$[0,t_0]$\u4e0a\u81f3\u5c11\u6709\u4e00\u4e2a\u96f6\u70b9.
                              \n<\/ol>
                              \n3. \u7b49\u8ddd\u3001Jacobi\u573a\u4e0e\u51fd\u6570\u7684Laplace<\/a><\/span>\n\n
                              1. \u5047\u8bbe$M$\u4e0e$N$\u662f\u9ece\u66fc\u6d41\u5f62, $\\phi: M\\to N$\u662f\u5c40\u90e8\u7b49\u8ddd\u6620\u5c04, \u4e14$N$\u662f\u5b8c\u5907\u7684. \u95ee:
                                \n
                                1. $M$\u662f\u5426\u4e5f\u5b8c\u5907?
                                  \n <\/li>
                                2. $\\phi$\u662f\u8986\u76d6\u6620\u5c04\u5417?
                                  \n <\/li><\/ol><\/li>
                                3. \u5047\u8bbe$x\\in M$, \u800c\u4e14\u4ece$x$\u51fa\u53d1\u7684\u6d4b\u5730\u7ebf$\\gamma$\u4e0a\u6240\u4ee5\u6b63\u5e38Jacobi\u573a\u90fd\u662f\u51e0\u4e4e\u5e73\u884c\u7684, \u5373\u5b58\u5728\u6cbf\u7740$\\gamma$\u7684\u5e73\u884c\u5355\u4f4d\u5411\u91cf\u573a$W(t)$, \u4f7f\u5f97\u6ee1\u8db3$U(0)=0$\u7684Jacobi\u573a$U(t)=f(t)W(t)$, \u5176\u4e2d$f(t)$\u662f\u5b9a\u4e49\u5728$[0,b]$\u4e0a\u7684\u5149\u6ed1\u51fd\u6570, $\\gamma:[0,b]\\to M$\u662f\u6d4b\u5730\u7ebf.
                                  \n
                                  1. \u4ee4$\\tilde{S}$\u662f$M_x$\u7684\u5b50\u7a7a\u95f4, \u5728$x$\u7684\u4e00\u4e2a\u90bb\u57df\u5185, \u4ee4$S=\\exp_x\\tilde{S}$. \u8bc1\u660e: \u5982\u679c$\\gamma'(0)\\in \\tilde{S}$, $\\gamma([0,b])\\subset S$, \u5219$P^\\gamma$\u5c06$\\tilde{S}$\u5e73\u884c\u79fb\u52a8\u5230$\\gamma(b)$\u7684\u5207\u7a7a\u95f4$S_{\\gamma(b)}$;
                                    \n <\/li>
                                  2. $M_x$\u4e2d\u6240\u6709\u622a\u9762\u66f2\u7387\u662f\u5e38\u6570;
                                    \n <\/li>
                                  3. \u5e38\u66f2\u7387\u7a7a\u95f4\u7684Jacobi\u573a\u90fd\u662f\u51e0\u4e4e\u5e73\u884c\u7684.
                                    \n <\/li><\/ol><\/li>
                                  4. \u5047\u8bbe$X,Y$\u662f\u6cbf\u7740$\\gamma:[0,b]\\to M$\u7684Jacobi\u573a, \u5219\u6709
                                    \n \\[
                                    \n \\langle X, D_{\\gamma’}Y \\rangle-\\langle D_{\\gamma’}X, Y \\rangle=\\mathrm{const}.
                                    \n \\]<\/li>
                                  5. \u5047\u8bbe$(M,g)$\u662f\u5177\u6709\u5e38\u8d1f\u622a\u9762\u66f2\u7387$c$\u7684\u9ece\u66fc\u6d41\u5f62, $\\gamma:[0,l]\\to M$\u662f\u6b63\u89c4\u6d4b\u5730\u7ebf, $v\\in M_{\\gamma(l)}$\u6ee1\u8db3$\\langle v,\\gamma'(l) \\rangle=0$, $\\lvert v \\rvert=1$. \u8bc1\u660e: \u6cbf\u7740$\\gamma$\u6ee1\u8db3$J(0)=0$, $J(l)=v$\u7684Jacobi\u573a\u7531\u4e0b\u5f0f\u7ed9\u51fa:
                                    \n \\[
                                    \n J(t)=\\frac{\\sinh(\\sqrt{-c}t)}{\\sinh(\\sqrt{-c}l)}W(t),
                                    \n \\]
                                    \n \u5176\u4e2d$W(t)$\u662f\u6cbf\u7740$\\gamma$\u5e73\u884c\u7684\u5411\u91cf\u573a, \u4e14$W(0)=u_0\/u_0$, \u8fd9\u91cc$u_0=\\left\\{ (\\exp_{\\gamma(0)})_{*}l\\gamma'(0) \\right\\}^{-1}(v)$.<\/li>
                                  6. \u5047\u8bbe$(M,g)$\u662f\u9ece\u66fc\u6d41\u5f62, \u7ed9\u5b9a$O\\in M$, $\\rho$\u662f\u76f8\u5bf9\u4e8e$O$\u7684\u8ddd\u79bb\u51fd\u6570.
                                    \n
                                    1. $\\rho(x)$\u5728$O$\u9644\u8fd1\u4e0d\u662f$C^1$\u7684;
                                      \n <\/li>
                                    2. \u82e5$M$\u4e3a\u7d27\u6d41\u5f62, \u5219$\\rho(x)$\u5728$M\\setminus\\left\\{ O \\right\\}$\u4e0a\u4e5f\u4e0d\u662f$C^1$\u7684.
                                      \n <\/li><\/ol><\/li>
                                    3. \u5047\u8bbe$M$\u662f\u5177\u6709\u6b63\u622a\u9762\u66f2\u7387\u7684\u5947\u6570\u7ef4\u7d27\u9ece\u66fc\u6d41\u5f62, \u5219$M$\u662f\u4e0d\u53ef\u5b9a\u5411\u6d41\u5f62.<\/li>
                                    4. \u5047\u8bbe$(M,g)$\u662f\u5355\u8fde\u901a\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62, \u5bf9\u4efb\u610f\u7684$P\\in M$, \u82e5$P$\u70b9\u6cbf\u6240\u6709\u4ece$P$\u51fa\u53d1\u7684\u5f84\u5411\u6d4b\u5730\u7ebf\u7684\u7b2c\u4e00\u5171\u8f6d\u70b9\u90fd\u662f\u540c\u4e00\u70b9$Q$\u4e14$Q\\neq P$, $d(P,Q)=\\pi$. \u8bc1\u660e:\u5982\u679c$K_M\\leq 1$,\u5219$M$\u4e0e$S^n$\u7b49\u8ddd.<\/li>
                                    5. \u5047\u8bbe$\\gamma:[a,b]\\to M$\u662f\u6d4b\u5730\u7ebf, \u4e14$\\gamma(b)$\u4e0d\u662f$\\gamma(a)$\u7684\u5171\u8f6d\u70b9, \u5219\u5bf9\u4efb\u610f\u7684$x\\in M_{\\gamma(a)}$, $Y\\in M_{\\gamma(b)}$, \u5b58\u5728\u552f\u4e00\u6cbf\u7740$\\gamma$\u7684Jacobi\u573a$J(t)$\u6ee1\u8db3$J(a)=X$, $J(b)=Y$.<\/li>
                                    6. \u5047\u8bbe$\\gamma:[0,b]\\to M$\u662f\u6b63\u89c4\u6d4b\u5730\u7ebf, $\\gamma$\u4e0a\u65e0$\\gamma(0)$\u7684\u5171\u8f6d\u70b9. $J(t)$\u662f\u6ee1\u8db3$J(0)=0$, $\\lvert J'(0) \\rvert=1$\u7684\u6b63\u5e38Jacobi\u573a, \u5982\u679c\u5bf9\u4efb\u610f\u7684$v\\in M_{\\gamma(t)}$, \u5f53$v\\perp \\gamma'(t)$\u65f6, \u90fd\u6709$K(v,\\gamma'(t))\\leq\\beta$, \u5219
                                      \n \\[
                                      \n J(t)=
                                      \n \\begin{cases}
                                      \n \\frac{\\sin\\sqrt{\\beta}t}{\\sqrt{\\beta}},&\\beta>0,\\,t<\\frac{\\pi}{\\sqrt{\\beta}},\\\\\n t,&\\beta=0,\\\\\n \\frac{\\sinh\\sqrt{-\\beta}t}{\\sqrt{-\\beta}},&\\beta<0.\n \\end{cases}\n \\]\n<\/li>
                                    7. \u5047\u8bbe$M$\u662f\u4e8c\u7ef4\u8fde\u901a\u9ece\u66fc\u6d41\u5f62, $K$\u662fGauss\u66f2\u7387, \u4e14$K\\leq -c^2<0$. \u53c8\u8bbe$O\\in M$, $\\rho$\u662f\u5230$O$\u7684\u8ddd\u79bb\u51fd\u6570. \u8bc1\u660e: $(\\tanh(c\\rho\/2))^2$\u662f\u5149\u6ed1\u51fd\u6570, \u4e14$\\Delta(\\tanh(c\\rho\/2))^2>0$, \u8fd9\u91cc\u5047\u8bbe$c>0$.<\/li><\/ol>
                                      \n 4. \u5272\u8ff9\u3001\u5171\u8f6d\u70b9\u3001\u6307\u6570\u4e0e\u6d41\u5f62\u4e0a\u7684\u5fae\u5206\u7b97\u5b50<\/a><\/span>\n\n
                                      1. \u8bbe$B(\\delta)$\u662f$M$\u4e2d\u7684\u5f00\u7403, $B_\\delta=\\exp_x(B(\\delta)$, $x\\in M$, \u63cf\u8ff0\u7403\u9762S^n(1), \u534a\u5f84\u4e3a$1$\u7684\u5706\u67f1\u9762\u3001$\\mathbb{R}^n$\u53ca\u7403\u9762\u4e0a\u63a5\u6839\u7ec6\u7ba1\u540e\u7684\u66f2\u9762\u7684$B_\\delta$($\\delta>0$). \u5e76\u63cf\u8ff0\u8fd9\u4e9b\u7a7a\u95f4\u7684\u5171\u8f6d\u8f68\u8ff9\u4e0e\u5272\u8ff9.
                                        \n <\/li>
                                      2. \u7ed9\u51fa\u4e00\u4e2a\u7d27\u81f4\u9ece\u66fc\u6d41\u5f62\u7684\u4f8b\u5b50, \u4f7f\u5f97\u67d0\u4e2a$x\\in M$, $M_x$\u4e2d\u7684\u5207\u5272\u8ff9\u548c\u7b2c\u4e00\u5171\u8f6d\u8f68\u8ff9\u5747\u975e\u7a7a, \u4f46\u5b83\u4eec\u4e92\u4e0d\u76f8\u540c.
                                        \n <\/li>
                                      3. \u5047\u8bbe$M$\u662f\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62, $x\\in M$, \u4e14\u6709\u4ece$x$\u51fa\u53d1\u7684\u5c04\u7ebf, \u5219$M$\u5fc5\u987b\u662f\u975e\u7d27\u6d41\u5f62.
                                        \n <\/li>
                                      4. \u5047\u8bbe$M$\u4e0e$\\tilde{M}$\u90fd\u662f$n$\u4e3a\u9ece\u66fc\u6d41\u5f62, $\\gamma:[0,b]\\to M$\u4e0e$\\tilde{\\gamma}:[0,b]\\to\\tilde{M}$\u662f\u6b63\u89c4\u6d4b\u5730\u7ebf. \u4ee4$K(t), \\tilde{K}(t)$\u5206\u522b\u662fRauch\u6bd4\u8f83\u5b9a\u7406\u4e2d\u5b9a\u4e49\u7684\u51fd\u6570, \u4e14$\\tilde{K}(t)\\leq K(t)$\u5bf9\u4efb\u610f\u7684$t\\in[0,b]$\u90fd\u6210\u7acb. \u8bc1\u660e: \u6cbf\u6d4b\u5730\u7ebf$\\gamma$\u7684\u6307\u6570\u4e0d\u5c0f\u4e8e\u6cbf\u6d4b\u5730\u7ebf$\\tilde{\\gamma}$\u7684\u6307\u6570.
                                        \n <\/li>
                                      5. \u5047\u8bbe$M(c)$\u662f$n$\u7ef4\u7a7a\u95f4\u5f62\u5f0f, \u56fa\u5b9a$O\\in M(c)$, \u5efa\u7acb\u6d4b\u5730\u6781\u5750\u6807\u7cfb$(r,\\theta)$, \u4f7f\u9ece\u66fc\u5ea6\u91cf$g=dr^2+f^2(r)d\\sigma^2$, \u5176\u4e2d$d\\sigma^2$\u662f\u7403\u9762$S^{n-1}$\u4e2d\u7684\u6807\u51c6\u5ea6\u91cf$d\\sigma^2=h_{ij}(\\theta)d\\theta^i\\wedge d\\theta^j$, $1\\leq i, j\\leq n-1$. \u5f84\u5411\u51fd\u6570
                                        \n \\[
                                        \n f(r)=
                                        \n \\begin{cases}
                                        \n\t\\sin(\\sqrt{c}r)\/\\sqrt{c},&c>0,\\,r<\\pi\/\\sqrt{c},\\\\\n\tr,&c=0,\\\\\n\t\\sinh(\\sqrt{-c}r)\/\\sqrt{-c},&c<0.\n \\end{cases}\n \\]\n \u53c8\u8bbe$\\gamma:[0,b]\\to M(c)$\u662f\u4ece$O$\u51fa\u53d1\u7684\u6b63\u89c4\u6d4b\u5730\u7ebf, \u5176\u4e0a\u4e0d\u542b$O$\u7684\u5171\u8f6d\u70b9; $\\left\\{ e_1,\\ldots, e_{n-1},\\gamma'(0) \\right\\}$\u662f$M_O$\u7684\u5e7a\u6b63\u57fa, \u6cbf\u7740$\\gamma(t)$\u7684\u5e73\u884c\u79fb\u52a8\u5f97\u5230\u5c40\u90e8\u6807\u67b6\u573a$\\left\\{ e_1(t),\\ldots, e_{n-1}(t),\\gamma'(t) \\right\\}$; \u6cbf\u7740$\\gamma(t)$\u6784\u9020$n-1$\u4e2a\u6b63\u5e38\u7684\u76f8\u4e92\u6b63\u4ea4\u7684Jacobi\u573a$J_{i}(t)$, \u4f7f\u5f97$J_i(0)=0$, $J_i'(0)=e_i$, $i=1,2,\\ldots, n-1$.\n
                                          \n
                                        1. \u8bc1\u660e: $M(c)$\u7684\u4f53\u79ef\u5143\u4e3a$f^{n-1}(r)dr\\wedge dS^{n-1}$;\n <\/li>
                                        2. \u6c42$\\mathbb{R}^m$\u4e0e$H^m(-1)$\u4e2d\u534a\u5f84\u4e3a$R$\u7684\u6d4b\u5730\u7403\u7684\u4f53\u79ef.\n <\/li><\/ol>\n
                                        3. \u5047\u8bbe$\\Delta$\u4e0e$\\tilde{D}$\u5206\u522b\u662f$S^n$\u548c$\\mathbb{R}^{n+1}$\u4e2d\u5173\u4e8e\u6807\u51c6\u5ea6\u91cf\u7684Beltrami--Laplace\u7b97\u5b50. \u82e5$(x^1,\\ldots,x^{n+1})$\u662f$\\mathbb{R}^{n+1}$\u4e2d\u7684\u76f4\u89d2\u5750\u6807\u7cfb, \u4ee4$r=\\sqrt{\\sum_{i=1}^{n+1}(x^i)^2}$, \u5219$\\mathbb{R}^{n+1}$\u4e0a\u7684\u5149\u6ed1\u51fd\u6570\u53ef\u4ee5\u8868\u793a\u4e3a$f=f(x^1,\\ldots,x^{n+1})=f(r\\cdot\\vec p)$, \u5176\u4e2d$\\vec p\\in S^n$. \u8bc1\u660e\u5f53$r>0$\u65f6,
                                          \n \\[
                                          \n \\tilde{\\Delta}f=\\frac{\\partial^2f}{\\partial r^2}+\\frac{n}{r}\\frac{\\partial f}{\\partial r}+\\frac{1}{r^2}\\Delta f,
                                          \n \\]
                                          \n \u5176\u4e2d$\\Delta f$\u662f\u628a$f=f(r\\cdot\\vec p)$\u770b\u4f5c\u662f$\\vec p\\in S^n$, $r$\u56fa\u5b9a\u7684\u51fd\u6570\u5173\u4e8e$S^n$\u7684Laplace\u7b97\u5b50.
                                          \n <\/li>
                                        4. \u5047\u8bbe$M$\u662f\u7d27\u81f4\u5b9a\u5411\u9ece\u66fc\u6d41\u5f62, $f\\in C^\\infty(M)$, \u4e14$f\\geq0$, $\\Delta f\\geq0$ (\u8fd9\u6837\u7684$f$\u79f0\u4e3a\u975e\u8d1f\u6b21\u8c03\u548c\u51fd\u6570, \u5176\u4e2d$\\Delta$\u662fBeltrami–Laplace\u7b97\u5b50), \u5219$f$\u5fc5\u4e3a\u5e38\u6570.
                                          \n <\/li>
                                        5. \u5047\u8bbe$(M,g)$\u662f\u53ef\u5b9a\u5411\u9ece\u66fc\u6d41\u5f62, $\\phi,\\psi\\in A^r(M)$. \u8bc1\u660e:
                                          \n
                                          1. $\\langle \\phi,\\psi \\rangle=\\frac{1}{r!}\\phi^{i_1\\cdots i_r}\\psi_{j_1\\cdots j_r}$\u4e0e\u5c40\u90e8\u5750\u6807\u7cfb\u7684\u9009\u62e9\u65e0\u5173; \u4ece\u800c\u662f\u5927\u8303\u56f4\u5b9a\u4e49\u7684\u8fd0\u7b97;
                                            \n <\/li>
                                          2. \u661f\u7b97\u5b50$*\\phi$\u4e0e\u5b9a\u5411\u76f8\u7b26\u7684\u5c40\u90e8\u5750\u6807\u7cfb\u9009\u53d6\u65e0\u5173, \u4e5f\u662f\u6d41\u5f62\u4e0a\u5927\u8303\u56f4\u5b9a\u4e49\u7684\u8fd0\u7b97.
                                            \n <\/li><\/ol>
                                          3. \u5728\u9ece\u66fc\u6d41\u5f62$(M,g)$\u4e0a, $f\\in C^\\infty(M)$. \u6c42$\\Delta(\\lvert \\nabla f \\rvert^2)$, \u5e76\u8bc1\u660e:
                                            \n
                                            1. \u82e5$M$\u662f\u7d27\u81f4\u65e0\u8fb9\u7684, $\\mathrm{Ric}(M)\\geq0$, $\\Delta f=\\mathrm{const}.$ \u5219$\\nabla f$\u662f\u5e73\u884c\u5411\u91cf\u573a;
                                              \n <\/li>
                                            2. \u82e5$\\mathrm{Ric}(M)\\geq0$, $\\Delta f=\\mathrm{const}.$, $\\lvert \\nabla f \\rvert=\\mathrm{const}$.. \u5219$\\nabla f$\u662f\u5e73\u884c\u5411\u91cf\u573a.
                                              \n <\/li><\/ol>
                                            3. \u5982\u679c$M$\u4e0e$\\tilde{M}$\u662f\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62, $\\pi: \\tilde{ M}\\to M$\u662f\u5c40\u90e8\u7b49\u8ddd\u6620\u5c04, \u53c8\u662f$k$\u91cd\u8986\u76d6\u6620\u5c04. \u8bc1\u660e: $V(\\tilde{ M})=kV(M)$.<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

                                              \\begin{document} \u6458\u8981. \u8fd9\u662f\u5b66\u4e60\u9ece\u66fc\u51e0\u4f55\u7684\u4e00\u4e9b\u601d\u8003\u9898, \u4e3b\u8981\u53c2\u8003\u4e66\u76ee\u4e3a\u4f0d\u9e3f\u7199\u7b49\u7f16\u8457\u7684\u300a\u9ece\u66fc\u51e0… \u7ee7\u7eed\u9605\u8bfb\u5173\u4e8e\u9ece\u66fc\u51e0\u4f55\u7684\u4e00\u4e9b\u601d\u8003\u9898<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[146,126],"class_list":["post-769","post","type-post","status-publish","format-standard","hentry","category-math","tag-sikaoti","tag-limanjihe","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/769","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=769"}],"version-history":[{"count":30,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/769\/revisions"}],"predecessor-version":[{"id":802,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/769\/revisions\/802"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=769"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=769"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=769"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}