Ngô Quốc Anh

December 15, 2009

R-G: Codazzi equations in classical differential geometry

Filed under: Riemannian geometry — Ngô Quốc Anh @ 21:49

In classical differential geometry of surfaces, the Codazzi-Mainardi equations are expressed via the second fundamental form (L, M, N)

\displaystyle\begin{gathered}{L_v} - {M_u} = L\Gamma _{12}^1 + M(\Gamma _{12}^2 - \Gamma _{11}^1) - N\Gamma _{11}^2, \hfill \\{M_v} - {N_u} = L\Gamma _{22}^1 + M(\Gamma _{22}^2 - \Gamma _{12}^1) - N\Gamma _{12}^2. \hfill \\\end{gathered}

Consider a parametric surface in Euclidean space,

\displaystyle\mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v))

where the three component functions depend smoothly on ordered pairs (u,v) in some open domain U in the uv-plane. Assume that this surface is regular, meaning that the vectors \mathbf{r}_u and \mathbf{r}_v are linearly independent. Complete this to a basis \{\mathbf{r}_u,\mathbf{r}_v,\mathbf{n}\}, by selecting a unit vector \mathbf{n} normal to the surface. The unit vector \mathbf{n} is nothing but

\displaystyle\mathbf{n}=\frac{\mathbf{r}_u \times \mathbf{r}_v}{\|\mathbf{r}_u \times \mathbf{r}_v\|}.

It is possible to express the second partial derivatives of \mathbf{r} using the Christoffel symbols and the second fundamental form.

\displaystyle\begin{gathered}{{\mathbf{r}}_{uu}} = \Gamma _{11}^1{{\mathbf{r}}_u} + \Gamma _{11}^2{{\mathbf{r}}_v} + L{\mathbf{n}}, \hfill \\{{\mathbf{r}}_{uv}} = \Gamma _{12}^1{{\mathbf{r}}_u} + \Gamma _{12}^2{{\mathbf{r}}_v} + M{\mathbf{n}}, \hfill \\{{\mathbf{r}}_{vv}} = \Gamma _{22}^1{{\mathbf{r}}_u} + \Gamma _{22}^2{{\mathbf{r}}_v} + N{\mathbf{n}}. \hfill \\ \end{gathered}

Clairaut’s theorem states that partial derivatives commute


If we differentiate \mathbf{r}_{uu} with respect to v and \mathbf{r}_{uv} with respect to u, we get

\displaystyle \begin{gathered}{\left( {\Gamma _{11}^1} \right)_v}{{\mathbf{r}}_u} + \Gamma _{11}^1{{\mathbf{r}}_{uv}} + {\left( {\Gamma _{11}^2} \right)_v}{{\mathbf{r}}_v} + \Gamma _{11}^2{{\mathbf{r}}_{vv}} + {L_v}{\mathbf{n}} + L{{\mathbf{n}}_v} \hfill \\ \qquad = {\left( {\Gamma _{12}^1} \right)_u}{{\mathbf{r}}_u} + \Gamma _{12}^1{{\mathbf{r}}_{uu}} + {\left( {\Gamma _{12}^2} \right)_u}{{\mathbf{r}}_v} + \Gamma _{12}^2{{\mathbf{r}}_{uv}} + {M_u}{\mathbf{n}} + M{{\mathbf{n}}_u} \hfill \\ \end{gathered}

Now substitute the above expressions for the second derivatives and equate the coefficients of \mathbf{n}

\displaystyle M \Gamma_{11}^1 + N \Gamma_{11}^2 + L_v = L \Gamma_{12}^1 + M \Gamma_{12}^2 + M_u

Rearranging this equation gives the first Codazzi equation. The second equation may be derived similarly.



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