Answer to Question #144905 in Differential Geometry | Topology for anjali

Question #144905
The third fundamental form of a surface σ(u, v) is
||N̂u|| ^2 du^2 + 2N̂u.N̂v dudv + ||N̂v||^2 dv^2
where N̂ (u, v) is the standard unit normal to σ(u, v). Let FIII be the associated 2 × 2
symmetric matrix.
Show that FIII = FIIF^−1I FII , where FI and FII are the 2 × 2 symmetric matrices
associated with the first and the second fundamental forms, respectively
1
Expert's answer
2020-11-22T16:39:23-0500

"I = dS^2=a_{11}(du^1)^2+2a_{12}du^1du^2+a_{22}(du^2)^2\\\\\na_{ij}=(\\overline{r_i}, \\overline{r_j}); i,j=1,2\\\\\nII=b_{11}(du^1)^2+2b_{12}du^1du^2+b_{22}(du^2)^2\\\\\nb_{ij}=(\\overline{r_{ij}},\\overline{n}); i,j=1,2\\\\\nIII=d\\overline{n}^2=(\\overline{n_{u^1}})^2(du^1)^2+2\\overline{n_{u^1}}\\overline{n_{u^2}}du^1du^2+(\\overline{n_{u^2}})^2(du^2)^2\\\\\n\\begin{pmatrix}\n (\\overline{n_{u^1}})^2 & \\overline{n_{u^1}}\\overline{n_{u^2}} \\\\\n \\overline{n_{u^1}}\\overline{n_{u^2}} & (\\overline{n_{u^2}})^2 \n\\end{pmatrix}\n= \n\\begin{pmatrix}\n b_{11} & b_{12} \\\\\n b_{12} & b_{22} \n\\end{pmatrix}\n\\cdot\n\\begin{pmatrix}\n a_{11} & a_{12} \\\\\n a_{12} & a_{22}\n\\end{pmatrix}^{-1}\n\\cdot\n\\begin{pmatrix}\n b_{11} & b_{12} \\\\\n b_{12} & b_{22} \n\\end{pmatrix}\\\\\nA=\\begin{pmatrix}\n a_{11} & a_{12} \\\\\n a_{12} & a_{22}\n\\end{pmatrix}\\\\\nA^{-1}=\\begin{pmatrix}\n \\frac{a_{22}}{\\Delta} & -\\frac{a_{12}}{\\Delta} \\\\\n -\\frac{a_{12}}{\\Delta} & \\frac{a_{11}}{\\Delta}\n\\end{pmatrix}\\\\\n\\Delta=|A|=a_{11}a_{22}-a_{12}^2"


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