4. Curvatures
Geometric quantities, and this concerns in particular the curvatures which we introduce shortly, must be invariant w.r.t. regular changes of the parametrization but also w.r.t. changes of the orthonormal normal frames. Inner geometric properties of surfaces must not depend on the choice of an ONF.
To provide an method to transform one ONF into another ONF while conserving the orientation, we consider matrix-valued mappings \[ {\mathbf R}=(r_{\sigma\omega})_{\sigma,\omega=1,\ldots,n}\in C^{3+\alpha}(B,SO_n) \] with the special orthogonal group \( SO_n, \) and \( {\mathbf R} \) satisfies the properties \begin{align} & \sum_{\sigma=1}^nr_{\sigma\omega}(w)^2=\sum_{\sigma=1}^nr_{\omega\sigma}(w)^2=1\quad\text{for all}\ \omega=1,\ldots,n, \\[1ex] & \sum_{\sigma=1}^nr_{\sigma\omega}(w)r_{\sigma\omega'}(w)=\sum_{\sigma=1}^nr_{\omega\sigma}(w)r_{\omega'\sigma}(w)=0\quad\text{for all}\ \omega\not=\omega' \end{align} for all \( w\in B \) and, to preserve the orientation, \[ \det{\mathbf R}(w)=1\quad\text{in}\ B. \]
4.1.2 Rotation of orthogonal normal frames
Let \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF. The action of \( {\mathbf R} \) on \( {\mathfrak N} \) results in \( n \) new unit normal vectors \[ \overset{\boldsymbol{\,\sim}}{N}_\sigma(w):=\sum_{\omega=1}^nr_{\sigma\omega}(w)N_\omega(w),\quad\sigma=1,\ldots,n. \] We compute \begin{align} |\overset{\boldsymbol{\,\sim}}{N}_\sigma|^2 &= \langle\overset{\boldsymbol{\,\sim}}{N}_\sigma,\overset{\boldsymbol{\,\sim}}{N}_\sigma\rangle =\sum_{\omega=1}^n\sum_{\omega'=1}^n\langle r_{\sigma\omega}N_\omega,r_{\sigma\omega'}N_{\omega'}\rangle \\[1ex] &= \sum_{\omega=1}^n\sum_{\omega'=1}^nr_{\sigma\omega}r_{\sigma\omega'}\delta_{\omega\omega'} =\sum_{\omega=1}^nr_{\sigma\omega}^2 =1 \end{align} and, analoguously, \[ \langle\overset{\boldsymbol{\,\sim}}{N}_\sigma,\overset{\boldsymbol{\,\sim}}{N}_\vartheta\rangle =\sum_{\omega=1}^n\sum_{\omega'=1}^nr_{\sigma\omega}r_{\vartheta\omega'}\delta_{\omega\omega'} =0 \] for \( \sigma\not=\vartheta. \)
Corollary: The new frame \( \overset{\boldsymbol{\,\sim}}{{\mathfrak N}}=(\overset{\boldsymbol{\,\sim}}{N}_1,\ldots,\overset{\boldsymbol{\,\sim}}{N}_n) \) is again an orthonormal normal frame.
In case \( n=2 \) of two codimensions we use the rotation matrix \[ {\mathbf R}=\left(\begin{array}{cc} \cos\varphi & \sin\varphi \\ -\sin\varphi & \cos\varphi \end{array}\right). \] Starting with an ONF \( (N_1,N_2), \) we obtain the new ONF \[ \overset{\boldsymbol{\,\sim}}{N}_1=\cos\varphi N_1+\sin\varphi N_2\,,\quad \overset{\boldsymbol{\,\sim}}{N}_2=-\sin\varphi N_1+\cos\varphi N_2\,. \] For example, a rotation about the angle \( \varphi=\frac{\pi}{2} \) gives \[ \overset{\boldsymbol{\,\sim}}{N}_1=N_2\,,\quad \overset{\boldsymbol{\,\sim}}{N}_2=-N_1\,. \] In case \( n=3 \) of three codimensions the Euler rotation matrices \begin{align} {\mathbf R}_1 &:= \left(\begin{array}{ccc} \cos\varphi & \sin\varphi & 0 \\ -\sin\varphi & \cos\varphi & 0 \\ 0 & 0 & 1 \end{array}\right), \\[2ex] {\mathbf R}_2 &:= \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos\psi & \sin\psi & \\ 0 & -\sin\psi & \cos\psi \end{array}\right), \\[2ex] {\mathbf R}_3 &:= \left(\begin{array}{ccc} \cos\vartheta & \sin\vartheta & 0 \\ -\sin\vartheta & \cos\vartheta & 0 \\ 0 & 0 & 1 \end{array}\right) \end{align} may be used from which an effective rotation would arise in the form \[ {\mathbf R}={\mathbf R}_3\circ{\mathbf R}_2\circ{\mathbf R}_1\,, \] see i.e. → P. Funk 1962. However, we forego these matrices, and, instead, we will discuss general forms of rotations in case of three or more codimensions.
We begin with the
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization. The mean curvature \( H_N \) w.r.t. a unit normal vector \( N \) of \( X \) is defined as \[ H_N:=\frac{1}{2}\,\sum_{i,j=1}^2g^{ij}L_{N,ij}=\frac{L_{N,11}g_{22}-2L_{N,12}g_{12}+L_{N,22}g_{11}}{2W^2}\,. \]
Consider now an ONF \( {\mathfrak N}=(N_1,\ldots,N_n) \) of \( X, \) and let again \( H_\sigma:=H_{N_\sigma}. \)
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization and \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF of \( X. \) The mean curvature vector \( H \) of \( X \) is defined as \[ H:=\sum_{\sigma=1}^nH_\sigma N_\sigma\,. \]
A regular surface parametrization in \( \mathbb R^3 \) possesses, up to orientation, exactly one unit normal vector \( N \) and, therefore, exactly one mean curvature \( H_N \) or mean curvature vector \( H, \) respectively.
4.2.2 Transformation behaviour of the mean curvature vector
We again assume that the regular parameter transformation leaves any unit normal vectors of a given ONF unchanged, i.e. \[ \overset{\boldsymbol{\,\sim}}{N}(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =N(u(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}),v(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})). \]
Proposition: Let \( N \) be an unit normal vector. Then, it holds the transformation formula \[ \overset{\boldsymbol{\,\sim}}{H}_N=H_N\quad\mbox{in}\ B. \]
Proof: Taking the identities \[ \overset{\boldsymbol{\,\sim}}g{\ }^{\mspace{-0.3ex}ij} =\sum_{m,n=1}^2\overline\Lambda{\ }_{\mspace{-0.4ex}i}^{\mspace{-0.4ex}m}\overline\Lambda{\ }_{\mspace{-0.4ex}i}^{\mspace{-0.4ex}n}g^{ij}\,,\quad \overset{\boldsymbol{\!\sim}}L_{N,ij} =\sum_{r,s=1}^2\Lambda_i^r\Lambda_j^sL_{N,rs} \] from → this paragraph and from → this paragraph, respectively, into account, we compute \begin{align} \overset{\boldsymbol{\,\sim}}{H}_N &= \frac{1}{2}\,\sum_{i,j=1}^2\overset{\boldsymbol{\,\sim}}g{\ }^{\mspace{-0.3ex}ij}\overset{\boldsymbol{\!\sim}}L_{N,ij} \\[1ex] &= \frac{1}{2}\,\sum_{i,j=1}^2\sum_{m,n=1}^2\sum_{r,s=1}^2 \overline\Lambda{\ }_{\mspace{-0.4ex}m}^{\mspace{-0.4ex}i}\overline\Lambda{\ }_{\mspace{-0.4ex}n}^{\mspace{-0.4ex}j}\Lambda_i^r\Lambda_j^sg^{mn}L_{N,rs} \\[1ex] &= \frac{1}{2}\,\sum_{m,n=1}^2\sum_{r,s=1}^2\delta_m^r\delta_n^sg^{mn}L_{N,rs} \\[1ex] &= \frac{1}{2}\,\sum_{r,s=1}^2g^{rs}L_{N,rs} =H_N\,. \end{align} The proposition follows.\( \qquad\Box \)
We immediately obtain
Proposition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization and \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF of \( X. \) Furthermore, let \( \psi \) be a \( C^{4+\alpha} \)-regular parameter transformation which leaves the unit normal vectors \( N_\sigma \) unchanges. Then, the mean curvature vector \( H \) is invariant w.r.t. \( \psi, \) i.e. it holds the \[ \overset{\boldsymbol{\,\sim}}{H}=H. \]
4.2.3 ONF invariance of the mean curvature vector
The mean curvature vector is invariant w.r.t. the choice of the ONF of the surface. This is the contents of the
Proposition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization, and let \( {\mathfrak N}=(N_1,\ldots,N_n) \) and \( \overset{\boldsymbol{\,\sim}}{{\mathfrak N}}=(\overset{\boldsymbol{\,\sim}}{N}_1,\ldots,\overset{\boldsymbol{\,\sim}}{N}_n) \) be two ONF of \( X \) such that \[ \overset{\boldsymbol{\,\sim}}{N}_\sigma(w):=\sum_{\omega=1}^nr_{\sigma\omega}(w)N_\omega(w),\quad\sigma=1,\ldots,n. \] Then it holds \[ \overset{\boldsymbol{\,\sim}}{H}=H. \]
Proof: We compute \begin{align} \overset{\boldsymbol{\,\sim}}{H} &= \sum_{\sigma=1}^n\overset{\boldsymbol{\,\sim}}{H}_\sigma\overset{\boldsymbol{\,\sim}}{N}_\sigma =\frac{1}{2}\,\sum_{i,j=1}^2\sum_{\sigma=1}^ng^{ij}\overset{\boldsymbol{\,\sim}}{L}_{\sigma,ij}\overset{\boldsymbol{\,\sim}}{N}_\sigma =\frac{1}{2}\,\sum_{i,j=1}^2\sum_{\sigma=1}^ng^{ij}\langle X_{u^iu^j},\overset{\boldsymbol{\,\sim}}{N}_\sigma\rangle\overset{\boldsymbol{\,\sim}}{N}_\sigma \\[1ex] &= \frac{1}{2}\,\sum_{i,j=1}^2\sum_{\sigma=1}^n\sum_{\omega,\omega'=1}^n g^{ij}r_{\sigma\omega}r_{\sigma\omega'}\langle X_{u^iu^j},N_\omega\rangle N_{\omega'}\,. \end{align} Now we evaluate separately the sums for \( \omega\not=\omega' \) and \( \omega=\omega' \) and obtain \[ \begin{array}{lcl} \overset{\boldsymbol{\,\sim}}{H}\!\!\!\! & = & \displaystyle\!\!\!\! \frac{1}{2}\,\sum_{i,j=1}^2\sum_{\sigma=1}^n\sum_{{\omega,\omega'=1}\atop{\omega\not=\omega'}}^n g^{ij}r_{\sigma\omega}r_{\sigma\omega'}\langle X_{u^iu^j},N_\omega\rangle N_{\omega'} \\[2ex] & & \displaystyle\quad +\frac{1}{2}\,\sum_{i,j=1}^2\sum_{\sigma=1}^n\sum_{{\omega,\omega'=1}\atop{\omega=\omega'}}^n g^{ij}r_{\sigma\omega}r_{\sigma\omega'}\langle X_{u^iu^j},N_\omega\rangle N_{\omega'} \\[2ex] & = & \displaystyle\!\!\!\! \frac{1}{2}\,\sum_{i,j=1}^2\sum_{{\omega,\omega'=1}\atop{\omega\not=\omega'}}^n g^{ij}\left(\sum_{\sigma=1}^nr_{\sigma\omega}r_{\sigma\omega'}\right)\langle X_{u^iu^j},N_\omega\rangle N_{\omega'} \\[2ex] & & \displaystyle\quad +\frac{1}{2}\,\sum_{i,j=1}^2\sum_{\omega=1}^n g^{ij}\left(\sum_{\sigma=1}^nr_{\sigma\omega}^2\right)\langle X_{u^iu^j},N_\omega\rangle N_\omega \\[2ex] & = & \displaystyle\!\!\!\! \frac{1}{2}\,\sum_{i,j=1}^2\sum_{\omega=1}^ng^{ij}\langle X_{u^iu^j},N_\omega\rangle N_\omega =\frac{1}{2}\,\sum_{i,j=1}^2\sum_{\omega=1}^ng^{ij}L_{\omega,ij}N_\omega \\[3ex] & = & \displaystyle\!\!\!\! \sum_{\omega=1}^nH_\omega N_\omega =H. \end{array} \] This proves the proposition.\( \qquad\Box \)
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization. The Gauss curvature \( K_N \) w.r.t. a unit normal vector \( N \) of \( X \) is defined as \[ K_N:=\frac{L_{N,11}L_{N,22}-L_{N,12}^2}{g_{11}g_{22}-g_{12}^2}\,. \]
Consider now an ONF \( {\mathfrak N}=(N_1,\ldots,N_n) \) of \( X, \) and let \( K_\sigma:=K_{N_\sigma}. \)
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface parametrization and \( {\mathfrak N}=(N_1,\ldots,N_n) \) be an ONF of \( X. \) The Gauss curvature \( K \) of \( X \) is defined as \[ K:=\sum_{\sigma=1}^nK_\sigma\,. \]