\u5047\u8bbe$(M^m,g)\\hookrightarrow(\\mathbb R^n,\\bar g)$\u662f\u5d4c\u5165\u5230\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5b50\u6d41\u5f62. \u53c8\u8bbe\u5176\u5c40\u90e8\u53c2\u6570\u8868\u793a\u4e3a$X(u^1,\\ldots,u^m)=\\bigl(x^1(u^1,\\ldots,u^m),\\ldots,x^n(u^1,\\ldots,u^m)\\bigr)$. \u6709\u4ee5\u4e0b\u4e24\u4ef6\u4e8b\u60c5:<\/p>\n
\u9996\u5148, $X$\u5728\u5c40\u90e8\u5750\u6807\u4e0b\u53ef\u89c6\u4e3a$M\\to\\mathbb R^n$\u7684\u4e00\u4e2a\u6620\u5c04, \u800c$(M,g)$\u7684\u8bf1\u5bfc\u5ea6\u91cf(\u56e0\u4e3a\u662f\u5d4c\u5165)\u5c31\u662f
\n$$
\ng=X^*(\\bar g)
\n$$
\n\u5373
\n\\begin{align*}
\ng_{\\alpha\\beta}&=g\\left(\\frac{\\partial}{\\partial u^\\alpha},\\frac{\\partial}{\\partial u^\\beta}\\right)
\n=X^*(\\bar g)\\left(\\frac{\\partial}{\\partial u^\\alpha},\\frac{\\partial}{\\partial u^\\beta}\\right)
\n=\\bar g\\left(\\frac{\\partial X^i}{\\partial u^\\alpha}\\frac{\\partial}{\\partial x^i},\\frac{\\partial X^j}{\\partial u^\\beta}\\frac{\\partial }{\\partial x^j}\\right)\\\\
\n&=\\frac{\\partial X^i}{\\partial u^\\alpha}\\frac{\\partial X^j}{\\partial u^\\beta}\\delta_{ij}
\n=\\frac{\\partial X^i}{\\partial u^\\alpha}\\frac{\\partial X^i}{\\partial u^\\beta}.
\n\\end{align*}
\n\u8fd9\u5373\u662f\u6211\u4eec\u5e38\u8bf4\u7684\u7b2c\u4e00\u57fa\u672c\u578b.
\n<\/p>\n
\u5176\u6b21, \u82e5$f$\u662f\u5b9a\u4e49\u5728$M$\u4e0a\u7684\u4e00\u4e2a\u51fd\u6570, \u5219\u5728\u5c40\u90e8\u53c2\u6570\u5316\u4e0b,
\n$$
\n\\frac{\\partial}{\\partial u^\\alpha}.f:=\\frac{\\partial (f\\circ X)}{\\partial u^\\alpha}
\n=\\frac{\\partial f}{\\partial x^k}\\frac{\\partial X^k}{\\partial u^\\alpha}
\n=\\frac{\\partial X^k}{\\partial u^\\alpha}\\frac{\\partial}{\\partial x^k}.f.
\n$$
\n\u8fd9\u91cc\u9700\u8981\u6ce8\u610f, \u4e24\u4e2a$f$\u7684\u542b\u4e49\u7a0d\u6709\u4e0d\u540c, \u5728\u4e0a\u5f0f\u5b9a\u4e49\u7684\u5de6\u8fb9, $f$\u662f\u53ea\u5b9a\u4e49\u5728$M$\u4e0a\u7684\u51fd\u6570, \u4e00\u822c\u800c\u8a00\u5b83\u5728$\\mathbb R^n$\u4e2d\u4e0d\u4e00\u5b9a\u6709\u5b9a\u4e49. \u5728\u5b9a\u4e49\u7684\u53f3\u8fb9, \u6211\u4eec\u5c06$f$(\u4f8b\u5982\u5e38\u503c)\u5ef6\u62d3\u5230$M$\u5728$\\mathbb R^n$\u4e2d\u7684\u4e00\u4e2a\u90bb\u57df. \u4ece\u800c\u53ef\u4ee5\u5229\u7528\u5176\u5728$\\mathbb R^n$\u4e2d\u7684\u5bfc\u6570$\\frac{\\partial f}{\\partial x^k}$\u6765\u8ba1\u7b97\u5176\u5728\u6d41\u5f62\u4e0a\u7684\u5bfc\u6570.<\/p>\n
\u6700\u540e, \u6ce8\u610f\u5230$\\left\\{\\frac{\\partial}{\\partial x^k}\\right\\}_{k=1}^n$\u6070\u597d\u7ec4\u6210\u4e86$\\mathbb R^n$\u4e2d\u7684\u6807\u51c6\u6b63\u4ea4\u57fa, \u4e60\u60ef\u4e0a\u6211\u4eec\u5c06\u5176\u7b49\u540c\u4e8e0\u548c1\u7ec4\u6210\u7684\u6807\u51c6\u57fa. \u6545\u91c7\u7528\u5206\u91cf\u7684\u5199\u6cd5, \u6211\u4eec\u53ef\u7531\u4e0a\u9762\u8fd9\u79cd\u770b\u6cd5\u5f97\u5230
\n$$
\n\\frac{\\partial }{\\partial u^\\alpha}\\sim\\frac{\\partial X}{\\partial u^\\alpha}.
\n$$
\n\u8fd9\u5728\u5b50\u6d41\u5f62\u7684\u8ba1\u7b97\u4e2d\u662f\u4e00\u4e2a\u5e38\u7528\u7684\u7ea6\u5b9a. \u6211\u4eec\u5728\u5177\u4f53\u7684\u89e3\u8bfb\u65f6\u5b8c\u5168\u53ef\u4ee5\u6309\u7167\u4e0a\u9762\u7684\u5b9a\u4e49\u6765\u5373\u53ef.<\/p>\n
\u7279\u522b\u5730,
\n$$
\n\\nabla f=g^{\\alpha\\beta}\\frac{\\partial f}{\\partial u^\\alpha}\\frac{\\partial}{\\partial u^\\beta}\\sim g^{\\alpha\\beta}\\frac{\\partial f}{\\partial u^\\alpha}\\frac{\\partial X}{\\partial u^\\beta},
\n$$
\n\u4ece\u800c
\n\\begin{align*}
\ng\\left(\\frac{\\partial }{\\partial u^\\gamma},\\nabla f\\right)
\n&\\sim \\bar g\\left(\\frac{\\partial X}{\\partial u^\\gamma}, g^{\\alpha\\beta}\\frac{\\partial f}{\\partial u^\\alpha}\\frac{\\partial X}{\\partial u^\\beta}\\right)\\\\
\n&=\\frac{\\partial X^i}{\\partial u^\\gamma}g^{\\alpha\\beta}\\frac{\\partial f}{\\partial u^\\alpha}\\frac{\\partial X^i}{\\partial u^\\beta}
\n=g_{\\gamma\\beta}g^{\\alpha\\beta}\\frac{\\partial f}{\\partial u^\\alpha}
\n=\\frac{\\partial f}{\\partial u^\\gamma}.
\n\\end{align*}<\/p>\n","protected":false},"excerpt":{"rendered":"