\u6ce8\u610f\u5230
\n\\begin{align*}
\n\\overline\\nabla_X\\overline\\nabla_YZ&=\\overline\\nabla_X(\\nabla_YZ+A(Y,Z))\\\\
\n&=\\nabla_X\\nabla_YZ+A(X,\\nabla_YZ)+\\nabla^\\perp_XA(Y,Z)-P(A(Y,Z);X).
\n\\end{align*}
\n\u56e0\u6b64, \u6309\u7167\u9ece\u66fc\u66f2\u7387\u5f20\u91cf\u7684\u5b9a\u4e49:
\n\\begin{align*}
\n\\bar R(X,Y,Z,W)&=\\left\\langle \\bar R(X,Y)Z,W\\right\\rangle=\\left\\langle\\overline\\nabla_X\\overline\\nabla_Y Z-\\overline\\nabla_Y\\overline\\nabla_XZ-\\overline\\nabla_{[X,Y]}Z,W\\right\\rangle\\\\
\n&=\\Big\\langle\\nabla_X\\nabla_YZ+A(X,\\nabla_YZ)+\\nabla^\\perp_XA(Y,Z)-P(A(Y,Z);X)\\\\
\n&\\qquad-\\nabla_Y\\nabla_XZ-A(Y,\\nabla_XZ)-\\nabla^\\perp_YA(X,Z)+P(A(X,Z);Y)\\\\
\n&\\qquad-\\nabla_{[X,Y]}Z-A([X,Y],Z),W\\Big\\rangle\\\\
\n&=\\Big\\langle R(X,Y)Z,P(A(X,Z);Y)-P(A(Y,Z);X)\\\\
\n&\\qquad +\\nabla^\\perp_XA(Y,Z)-A(Y,\\nabla_XZ)-A(\\nabla_XY,Z)\\\\
\n&\\qquad -\\left(\\nabla_Y^\\perp A(X,Z)-A(X,\\nabla_YZ)-A(\\nabla_YX,Z)\\right),W\\Big\\rangle.
\n\\end{align*}
\n\u6ce8\u610f\u5230$A$\u53ef\u89c6\u4e3a$TM\\otimes TM\\to T^\\perp M$\u7684\u5f20\u91cf, \u6545\u5229\u7528$\\nabla$\u4e0e\u6cd5\u8054\u7edc$\\nabla^\\perp$, \u53ef\u4ee5\u5b9a\u4e49\u5f20\u91cf\u4e1b$T^*M\\otimes T^*M\\otimes T^\\perp M$\u4e0a\u7684\u8054\u7edc:
\n$$
\n(\\widetilde\\nabla_X A)(Y,Z):=\\nabla^\\perp_XA(Y,Z)-A(\\nabla_XY,Z)-A(Y,\\nabla_XZ).
\n$$
\n\u56e0\u6b64,
\n\\begin{align*}
\n\\bar R(X,Y,Z,W)&=\\Big\\langle R(X,Y)Z,P(A(X,Z);Y)-P(A(Y,Z);X)\\\\
\n&\\qquad (\\widetilde\\nabla_XA)(Y,Z)-(\\widetilde\\nabla_YA)(X,Z),W\\Big\\rangle.
\n\\end{align*}
\n\u6ce8\u610f\u5230, \u6211\u4eec\u6709\u5982\u4e0b\u7684Weingarten\u65b9\u7a0b:
\n\\begin{equation}
\n\\left\\langle P(U;X),Y\\right\\rangle
\n=-\\left\\langle\\overline\\nabla_XU,Y\\right\\rangle
\n=\\left\\langle\\overline\\nabla_XY,U\\right\\rangle
\n=\\left\\langle A(X,Y),U\\right\\rangle.
\n\\end{equation}
\n\u56e0\u6b64, \u6211\u4eec\u5f97\u5230Gauss\u65b9\u7a0b:
\n\\begin{equation}
\n\\bar R(X,Y,Z,W)=R(X,Y,Z,W)+\\left\\langle A(X,Z),A(Y,W)\\right\\rangle-\\left\\langle A(X,W),A(Y,Z)\\right\\rangle.
\n\\end{equation}<\/p>\n
\u7c7b\u4f3c\u5730,
\n\\begin{align*}
\n\\bar R(X,Y,Z,U)&=\\Big\\langle R(X,Y)Z,P(A(X,Z);Y)-P(A(Y,Z);X)\\\\
\n&\\qquad (\\widetilde\\nabla_XA)(Y,Z)-(\\widetilde\\nabla_YA)(X,Z),U\\Big\\rangle\\\\
\n&=\\Big\\langle (\\widetilde\\nabla_XA)(Y,Z)-(\\widetilde\\nabla_YA)(X,Z),U\\Big\\rangle.
\n\\end{align*}
\n\u7531\u6b64\u5f97\u5230Codazzi\u65b9\u7a0b:
\n\\begin{equation}
\n\\bar R(X,Y,Z,U)=\\Big\\langle (\\widetilde\\nabla_XA)(Y,Z)-(\\widetilde\\nabla_YA)(X,Z),U\\Big\\rangle.
\n\\end{equation}<\/p>\n
\u5b8c\u5168\u7c7b\u4f3c\u5730, \u6ce8\u610f\u5230
\n\\begin{align*}
\n\\overline\\nabla_X\\overline\\nabla_YU&=\\overline\\nabla_X(\\nabla^\\perp_YU-P(U;Y))\\\\
\n&=\\nabla_X^\\perp\\nabla_Y^\\perp U-P(\\nabla^\\perp_YU;X)-\\nabla_XP(U;Y)-A(X,P(U;Y)).
\n\\end{align*}
\n\u82e5\u5b9a\u4e49
\n$$
\nR^\\perp(X,Y,U,V):=\\left\\langle R^\\perp(X,Y)U,V\\right\\rangle=\\nabla_X^\\perp\\nabla_Y^\\perp U-\\nabla_Y^\\perp\\nabla_X^\\perp U-\\nabla^\\perp_{[X,Y]}U,
\n$$
\n\u5219
\n\\begin{align*}
\n\\bar R(X,Y,U,V)&=\\left\\langle \\bar R(X,Y)U,V\\right\\rangle=\\left\\langle\\overline\\nabla_X\\overline\\nabla_Y U-\\overline\\nabla_Y\\overline\\nabla_XU-\\overline\\nabla_{[X,Y]}U,V\\right\\rangle\\\\
\n&=\\Big\\langle\\nabla_X^\\perp\\nabla_Y^\\perp U-P(\\nabla^\\perp_YU;X)-\\nabla_XP(U;Y)-A(X,P(U;Y))\\\\
\n&\\qquad-\\nabla_Y^\\perp\\nabla_X^\\perp U+P(\\nabla^\\perp_XU;Y)+\\nabla_YP(U;X)+A(Y,P(U;X))\\\\
\n&\\qquad-\\nabla^\\perp_{[X,Y]}U+P(U;[X,Y]),V \\Big\\rangle\\\\
\n&=\\Big\\langle R^\\perp(X,Y)U-A(X,P(U;Y))+A(Y,P(U;X)),V\\Big\\rangle.
\n\\end{align*}
\n\u4e3a\u8fdb\u4e00\u6b65\u7b80\u5316\u8bb0\u53f7, \u8bb0$P_U(Y):=P(U;Y)$, \u5e76\u5b9a\u4e49
\n$$
\n[P_U,P_V]X=P_UP_V(X)-P_VP_U(X).
\n$$
\n\u5219\u6709Weingarten\u65b9\u7a0b\u77e5
\n\\begin{align*}
\n\\left\\langle P_UX,Y\\right\\rangle&=\\left\\langle P(X;U),Y\\right\\rangle=\\left\\langle A(X,Y),U\\right\\rangle=\\left\\langle P_UY,X\\right\\rangle,\\\\
\n\\left\\langle P_VP_UX,Y\\right\\rangle&=\\left\\langle A(P_UX,Y),V \\right\\rangle=\\left\\langle X,P_UP_VY\\right\\rangle.
\n\\end{align*}
\n\u56e0\u6b64
\n\\begin{align*}
\n\\left\\langle A(X,P(U;Y))-A(Y,P(U;X)),V \\right\\rangle&=\\left\\langle A(X,P_UY)-A(Y,P_UX),V\\right\\rangle\\\\
\n&=\\left\\langle P_VP_UY,X\\right\\rangle-\\left\\langle X,P_UP_VY\\right\\rangle\\\\
\n&=-\\left\\langle [P_U,P_V]Y,X \\right\\rangle\\\\
\n&=\\left\\langle P_UP_VX,Y\\right\\rangle-\\left\\langle Y,P_VP_UX\\right\\rangle\\\\
\n&=\\left\\langle [P_U,P_V]X,Y \\right\\rangle.
\n\\end{align*}
\n\u6545, \u6211\u4eec\u5f97\u5230Ricci\u65b9\u7a0b:
\n\\begin{equation}
\n\\bar R(X,Y,U,V)=\\Big\\langle R^\\perp(X,Y)U-\\left\\langle [P_U,P_V]X,Y \\right\\rangle,V\\Big\\rangle.
\n\\end{equation}
\n<\/div>\n","protected":false},"excerpt":{"rendered":"