1<\/a>,p.33,38])\u7684\u66f2\u7387\u7b26\u53f7\u7ea6\u5b9a, \u8ba1\u7b97\u5f97\u5230
\n\\begin{align*}
\n \\Delta v&=\\mathrm{Hess}_v(e_i,e_i)=\\langle \\nabla_{e_i}\\nabla v,e_i \\rangle\\\\
\n &=\\langle \\nabla_{e_i}\\nabla_{\\nabla u} \\nabla u,e_i\\rangle\\\\
\n &=\\langle R(e_i,\\nabla u)\\nabla u,e_i\\rangle+\\langle \\nabla_{\\nabla u}\\nabla_{e_i} \\nabla u,e_i\\rangle+\\langle \\nabla_{[e_i,\\nabla u]}\\nabla u,e_i\\rangle\\\\
\n &=\\mathrm{Ric}(\\nabla u,\\nabla u)+\\nabla u\\langle \\nabla_{e_i}\\nabla u,e_i \\rangle -\\langle \\nabla_{e_i} \\nabla u, \\nabla_{\\nabla u}e_i \\rangle+\\langle \\nabla_{[e_i,\\nabla u]}\\nabla u,e_i\\rangle.
\n\\end{align*}
\n\u73b0\u5728, \u6ce8\u610f\u5230
\n\\begin{align*}
\n\\nabla u\\langle \\nabla_{e_i}\\nabla u,e_i \\rangle&=\\langle\\nabla u,\\nabla\\Delta u\\rangle,\\\\
\n\\langle \\nabla_{[e_i,\\nabla u]}\\nabla u,e_i\\rangle&=\\mathrm{Hess}_u([e_i,\\nabla u],e_i)=\\langle \\nabla_{e_i}\\nabla u,[e_i,\\nabla u] \\rangle\\\\
\n&=\\langle\\nabla_{e_i}\\nabla u,\\nabla_{e_i}\\nabla u-\\nabla_{\\nabla u}e_i\\rangle\\\\
\n&=\\left\\langle\\langle\\nabla_{e_i}\\nabla u,e_j\\rangle e_j,\\langle\\nabla_{e_i}\\nabla u,e_k\\rangle e_k\\right\\rangle-\\langle\\nabla_{e_i}\\nabla u,\\nabla_{\\nabla u}e_i\\rangle\\\\
\n&=|\\mathrm{Hess}_u|^2-\\langle\\nabla_{e_i}\\nabla u,\\nabla_{\\nabla u}e_i\\rangle.
\n\\end{align*}
\n\u56e0\u6b64
\n\\begin{equation}\\label{eq:Bochner-temp}
\n\\Delta v=\\mathrm{Ric}(\\nabla u,\\nabla u)+\\langle\\nabla u,\\nabla\\Delta u\\rangle+|\\mathrm{Hess}_u|^2-2\\langle\\nabla_{e_i}\\nabla u,\\nabla_{\\nabla u}e_i\\rangle.
\n\\end{equation}<\/p>\n\u4e0b\u9762, \u6211\u4eec\u5c06\u8bc1\u660e
\n$$
\n\\langle\\nabla_{e_i}\\nabla u,\\nabla_{\\nabla u}e_i\\rangle=0.
\n$$<\/p>\n
\u4e00\u79cd\u7b80\u5355\u7684\u65b9\u5f0f\u662f, \u6ce8\u610f\u5230, \u5b83\u4e0d\u4f9d\u8d56\u4e8e\u5750\u6807\u7684\u9009\u53d6(\u7531\\eqref{eq:Bochner-temp})\u3002\u53d6\u6cd5\u5750\u6807$\\{e_i\\}$, \u4f7f\u5f97$\\nabla_{e_i}e_j=0$ \u5728\u7ed9\u67d0\u7ed9\u5b9a\u70b9$p$\u5904\u6210\u7acb\u3002\u82e5\u8bbe$\\nabla u=\\langle\\nabla u,e_i\\rangle e_i=(e_iu)e_i:=u_ie_i$, \u5219\u5728$p$\u70b9\u5904
\n$$
\n\\nabla_{\\nabla u}e_i=\\nabla_{u_je_j}e_i=u_j\\nabla_{e_j}e_i=0.
\n$$<\/p>\n
\u53e6\u4e00\u79cd, \u4e0d\u5229\u7528\u6cd5\u5750\u6807\u7684\u529e\u6cd5\u662f\uff1a\u5047\u8bbe
\n$$
\n\\nabla_{e_i}\\nabla u=a_{ij}e_j,\\quad
\n\\nabla_{\\nabla u}e_i=b_{ij}e_j.
\n$$
\n\u5219
\n$$
\n\\langle\\nabla_{e_i}\\nabla u,\\nabla_{\\nabla u}e_i\\rangle=a_{ij}b_{ij}.
\n$$
\n\u6ce8\u610f\u5230,
\n\\begin{align*}
\na_{ij}&=\\langle\\nabla_{e_i}\\nabla u,e_j\\rangle=\\mathrm{Hess}_u(e_i,e_j)=a_{ji}\\\\
\nb_{ij}&=\\langle\\nabla_{\\nabla u}e_i,e_j\\rangle=\\nabla u\\langle e_i,e_j\\rangle-\\langle e_i,\\nabla_{\\nabla u}e_j\\rangle\\\\
\n&=-\\langle e_i, b_{jk}e_k\\rangle=-b_{ji}.
\n\\end{align*}
\n\u4e8e\u662f
\n$$
\n\\sum_{i,j}a_{ij}b_{ij}=\\sum_{j,i}a_{ji}b_{ji}=-\\sum_{j,i}a_{ij}b_{ij}.
\n$$
\n\u7531\u6b64\u5373\u5f97\u7ed3\u8bba\u3002
\n<\/div><\/p>\n
\u6700\u540e, \u4f5c\u4e3a\u6ce8\u8bb0, \u6211\u4eec\u5229\u7528\u5c40\u90e8\u5206\u91cf\u6765\u8ba1\u7b97, \u548c\u524d\u9762\u4e00\u6837\u5047\u8bbe$\\set{e_i}$\u662f$p$\u7684\u4e00\u4e2a\u90bb\u57df\u4e0a\u7684\u6cd5\u5750\u6807\u7cfb\u3002\u5b9a\u4e49
\n\\begin{align*}
\n\\nabla u&=\\inner{\\nabla u,e_i}e_i:=u_ie_i,\\\\
\nD^2u(e_i,e_j)&=\\inner{\\nabla_{e_i}\\nabla u,e_j}=\\inner{\\nabla_{e_i}(u_ke_k),e_j}\\\\
\n&=e_iu_k\\delta_{kj}=e_iu_j:=u_{ji}=u_{ij},\\\\
\n\\Delta u&=u_{ii},\\quad\\nabla\\Delta u=\\nabla u_{ii}=u_{iik}e_k,\\quad\\Delta u_k=u_{kii},\\\\
\n\\nabla_{e_i}\\nabla u&=\\nabla_{e_i}(u_ke_k)=u_{ki}e_k,\\\\
\n\\nabla_{\\nabla u}e_i&=\\nabla_{u_ke_k}e_i=0,\\\\
\n[e_i,\\nabla u]&=\\nabla_{e_i}\\nabla u-\\nabla_{\\nabla u}e_i=u_{ki}e_k,\\\\
\n\\nabla_{[e_i,\\nabla u]}\\nabla u&=\\nabla_{u_{ki}e_k}(u_le_l)=u_{ki}u_{lk}e_l,\\\\
\n\\nabla_{\\nabla u}\\nabla u&=\\nabla_{u_ke_k}(u_le_l)=u_ku_{lk}e_l,\\\\
\n\\nabla_{\\nabla u}(u_{ki}e_k)&=\\nabla_{\\nabla u}u_{ki}e_k=u_lu_{kil}e_k,\\\\
\n\\langle R(e_i,\\nabla u)\\nabla u,e_j\\rangle&=\\langle ([\\nabla_{e_i},\\nabla_{\\nabla u}]-\\nabla_{[e_i,\\nabla u]})\\nabla u,e_j\\rangle\\\\
\n&=\\langle \\nabla_{e_i}(u_ku_{lk}e_l)-\\nabla_{\\nabla u}(u_{ki}e_k)-u_{ki}u_{lk}e_l,e_j\\rangle\\\\
\n&=\\langle (u_{ki}u_{lk}+u_ku_{lki})e_l-u_lu_{kil}e_k-u_{ki}u_{lk}e_l,e_j\\rangle\\\\
\n&=u_ku_{lki}\\delta_{lj}-u_lu_{kil}\\delta_{kj}\\\\
\n&=u_ku_{jki}-u_lu_{jil}\\\\
\n&=u_k(u_{jki}-u_{jik})\\\\
\n\\mathrm{Ric}(\\nabla u,\\nabla u)&=\\mathrm{Ric}(u_ie_i,u_je_j)=u_iu_j\\mathrm{Ric}(e_i,e_j):=R_{ij}u_iu_j\\\\
\n\\end{align*}
\n\u53ef\u89c1
\n$$
\n\\langle R(e_i,\\nabla u)\\nabla u,e_i\\rangle=\\mathrm{Ric}(\\nabla u,\\nabla u)=
\nR_{ij}u_iu_j=u_j(u_{iji}-u_{iij})=u_j(u_{jii}-u_{iij})=u_ju_{jii}-\\inner{\\nabla u,\\nabla\\Delta u}
\n$$
\n\u5219
\n\\begin{align*}
\nv&=\\frac{1}{2}|\\nabla u|^2=\\frac{1}{2}u_k^2\\\\
\n\\Delta v&=D^2v(e_i,e_i)=v_{ii}=u_{ki}u_{ki}+u_ku_{kii}\\\\
\n&=|\\mathrm{Hess u}|^2+\\mathrm{Ric}(\\nabla u,\\nabla u)+\\inner{\\nabla u,\\nabla\\Delta u}.
\n\\end{align*}
\n
\u6ce8\u8bb0 1<\/span>.<\/span> \u4e0a\u8ff0\u63a8\u5bfc\u5b9e\u9645\u4e0a\u544a\u8bc9\u6211\u4eec\u5982\u4e0b\u7684Ricci\u6052\u7b49\u5f0f\uff1a
\n$$
\nu_{iki}-u_{iik}=u_{kii}-u_{iik}=R_{ik}u_i.
\n$$
\n<\/div><\/p>\n