2. Fundamental forms
We will often use the notation \[ u^1:=u,\quad u^2:=v \] for the parameters \( (u,v)\in B. \)
Definition: Let \( I(X)\in\mathbb R^{2\times 2} \) be a regular surface. Its first fundamental form \( I(X)\in\mathbb R^{2\times 2} \) is the symmetric matrix given by \[ I(X)=(g_{ij})_{i,j=1,2}\,,\quad g_{ij}:=\langle X_{u^i},X_{u^j}\rangle\,. \]
In particular, we write \begin{align} g_{11} &= \langle X_u,X_u\rangle\,, \\[0.6ex] g_{12} &= \langle X_u,v_v\rangle=\langle X_v,X_u\rangle=g_{21}\,, \\[0.6ex] g_{22} &= \langle X_v,X_v\rangle\,. \end{align}
The definition of the first fundamental form does not depend on the \( n. \) It rather is induced by the Euclidean metric of the embedding space \( \mathbb R^{n+2}. \) Namely, denote by \[ ds^2:=dx_1^2+\ldots+dx_{n+2}^2 \] the square of the standard line element \( ds \) of the space \( \mathbb R^{n+2} \) with its Euclidean coordinates \( x_1,\ldots,x_{n+2}. \) Embedding a regular surface into this space means analytically \begin{align} ds^2 &= (x_{1,u}\,du+x_{1,v}\,dv)^2+\ldots+(x_{n+2,u}\,du+x_{n+2,v}\,dv)^2 \\[0.6ex] &= \langle X_u,X_u\rangle\,du^2+2\langle X_u,X_v\rangle\,dudv+\langle X_v,X_v\rangle\,dv^2 \\[0.6ex] &= g_{11}\,du^2+2g_{12}\,dudv+g_{22}\,dv^2 \end{align} or, in a compact form, \[ ds^2=\sum_{i,j=1,}^2g_{ij}\,du^idu^j \] for the line element of the surface. Now, the area element \( W, \) defined → here, takes the form \[ W=\sqrt{g_{11}g_{22}-g_{12}^2}=\sqrt{\mbox{det}\,I(X)}\,, \] where \( \mbox{det}\,I(X)\gt 0 \) due to the regularity condition.
2.2.1 Definition of the second fundamental form
The second fundamental form \( I\!I(X) \) of a regular surface \( X\in C^{4+\alpha}(B,\mathbb R^3) \) is the symmetric matrix \[ I\!I(X)=(L_{ij})_{i,j=1,2}\,,\quad L_{ij}:=-\langle X_{u^i},N_{u^j}\rangle\,. \] To verify the symmetry we recall \[ \langle X_{u^i},N\rangle=0,\quad i=1,2, \] with the unit normal vector \( N \) of \( X. \) Differentiation yields \begin{align} 0 &= \langle X_{u^iu^j},N\rangle+\langle X_{u^i},N_{u^j}\rangle\,, \\[0.6ex] 0 &= \langle X_{u^ju^i},N\rangle+\langle X_{u^j},N_{u^i}\rangle \end{align} and due to \( X_{u^iu^j}=X_{u^ju^i} \) we obtain \( L_{ij}=L_{ji}. \) In case of higher codimension \( n\gt 1 \) it must be noted that there is a whole bundle of unit normal vectors instead of one \( N. \)
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface and let \( N\colon B\to\mathbb R^{n+2} \) be an unit normal vector of \( X. \) The second fundamental form of the surface \( X \) w.r.t. \( N \) is then defined by the symmetric matrix \[ I\!I_N(X)=(L_{N,ij})_{i,j=1,2}\,,\quad L_{N,ij}:=-\langle X_{u^i},N_{u^j}\rangle\,. \]
2.2.2 Definition of the third fundamental form
We complete the system of the first and the second fundamental forms by the following
Definition: Let \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) be a regular surface and let \( N\colon B\to\mathbb R^{n+2} \) be an unit normal vector of \( X. \) The third fundamental form of the surface \( X \) w.r.t. \( N \) is then defined by the symmetric matrix \[ I\!I\!I_N(X)=(e_{N,ij})_{i,j=1,2}\,,\quad e_{N,ij}:=\langle N_{u^i},N_{u^j}\rangle\,. \]
The symmetry of \( I\!I\!I\!_N(X) \) follows from \( \langle N_{u^i},N\rangle=0 \) for \( i=1,2 \) and, thus, \begin{align} 0 &= \langle N_{u^iu^j},N\rangle+\langle N_{u^i},N_{u^j}\rangle\,, \\[0.6ex] 0 &= \langle N_{u^ju^i},N\rangle+\langle N_{u^j},N_{u^i}\rangle\,, \end{align} together with \( N_{u^iu^j}=N_{u^ju^i}. \)
2.3.1 The length of a curve on a surface
We consider a regular curve \( c\in C^{4+\alpha}(I,B) \) on some interval \( I\subseteq\mathbb R, \) explicitly given by \[ c(t)=(u(t),v(t)),\quad t\in I, \] and its spatial image \( X\circ c\in\mathbb R^{n+2} \) on the regular surface \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}). \) Then, the length of this surface curve is given by (for reasons of simplicity we suppress variables) \begin{align} {\mathcal L}[X\circ c] =& \int\limits_I\left|\,\frac{d}{dt}\,X(u,v)\right|\,dt \\[1ex] =& \int\limits_I\sqrt{\langle X_u\frac{du}{dt}+X_v\frac{dv}{dt}\,,X_u\frac{du}{dt}+X_v\frac{dv}{dt}}\,dt \\[1ex] =& \int\limits_I\sqrt{\sum_{i,j=1}^2g_{ij}\,\frac{du^i}{dt}\frac{du^j}{dt}}\,dt. \end{align} Thus, the first fundamental form \( I(X) \) determines the length measurement on a surface.
2.3.2 The angle between two curves on a surface
Now we consider two regular curves \[ c_1(t)=(u_1(t),v_1(t)),\quad c_2(t)=(u_2(t),v_2(t)) \] of class \( C^{4+\alpha}(I,B) \) on some interval \( I, \) and their images \( k_1(t):=X\circ c_1(t) \) and \( k_2(t):=X\circ c_2(t) \) on the surface. Let \( t_0\in\mathring I \) be a point of intersection of the curves, i.e. \begin{align} & c_1(t_0)=c_2(t_0), \\[0.6ex] & c_1(t)\not=c_2(t)\quad\mbox{for all}\ t\in(t_0-\varepsilon,t_0+\varepsilon)\setminus\{t_0\}\,, \end{align} and let the surface \( X \) represent an embedding without self-intersections in a neighbourhood of the intersection point \( X\circ c_1(t_0)=X\circ c_2(t_0), \) where this neighbourhood contains also the image of \( (t_0-\varepsilon,t_0+\varepsilon). \) The tangential lines of the surface curves form an angle \( \alpha, \) given by (we mean the smaller of the two angles and omit the necessary computations) \begin{align} \cos\alpha &= \frac{\langle\dot k_1(t),\dot k_2(t)\rangle}{|\dot k_1(t)||\dot k_2(t)|} \\[1ex] &= \frac{g_{11}\dot u_1\dot u_2+g_{12}\,\{\dot u_1\dot v_2+\dot u_2\dot v_1\}+g_{22}\dot v_1\dot v_2} {\sqrt{g_{11}\dot u_1^2+2g_{12}\dot u_1\dot v_1+g_{22}\dot v_1^2}\,\sqrt{g_{11}\dot u_2^2+2g_{12}\dot u_2\dot v_2+g_{22}\dot v_2^2}}\,. \end{align} Thus, the first fundamental form \( I(X) \) also determines the intersection angles of surface curves.
Definition: The regular surface \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) is called conformally parametrized if the following conformality relations hold \[ \langle X_u,X_u\rangle=W=\langle X_v,X_v\rangle\,,\quad \langle X_u,X_v\rangle=0 \quad\mbox{in}\ B \] with the area element \( W. \)
Conformal coordinates diagonalize the line element \[ ds^2=g_{11}\,du^2+2g_{12}\,dudv+g_{22}\,dv^2=W(du^2+dv^2) \] on the whole disc \( B. \) Taking the following result of → S. Sauvigny 1999 into account we are entitled to apply such special coordinates due to our regularity assumptions.
Proposition: Assume that the coefficients \( a, \) \( b \) and \( c \) of the line element \[ ds^2=a\,du^2+2b\,dudv+c\,dv^2\,,\quad ac-b^2\gt 0\quad\mbox{in}\ B, \] are of class \( C^{1+\alpha}(B,\mathbb R) \) with \( \alpha\in(0,1). \) Then there is a conformal parameter system \( (u,v)\in B. \)
While Sauvigny's result holds in the large, stronger statements hold true in the small. For example, the following goes back to → S.-S. Chern 1955.
Proposition: Assume that the coefficients \( a, \) \( b \) and \( c \) of the line element \[ ds^2=a\,du^2+2b\,dudv+c\,dv^2\,,\quad ac-b^2\gt 0\quad\mbox{in}\ B, \] are Hölder continuous in \( B. \) Then for every point \( w\in\mathring B \) there exists an open neighbourhood over which the surface can be parametrized conformally.
2.4.2 Geodesic polar coordinates
For a detailed introduction of the exponential map, of geodesic discs and of geodesic polar coordinates we refer to introductory textbooks of differential geometry, for example → W. Blaschke and K. Leichtweiß 1973, → W. Klingenberg 2004, or → D. Laugwitz 1977. Here we just want to provide some important facts for later discussions.
Assume that the regular surface is given as a geodesic disc \( {\mathfrak B}_r(X_0) \) of geodesic radius \( r\gt 0 \) and with center \( X_0\in\mathbb R^{n+2}, \) or \( {\mathfrak B}_r(X_0) \) is a part of the surface. Using geodesic polar coordinates \( (\varrho,\varphi)\in[0,r]\times[0,2\pi], \) the surface can be written parametrically in the form \[ Z=Z(\varrho,\varphi)\colon[0,r]\times[0,2\pi]\longrightarrow\mathbb R^{n+2}\,. \] The new line element \( ds_P^2 \) reads \begin{align} ds_P^2 &= |Z_\varrho^2|^2\,d\varrho^2+2\langle Z_\varrho,Z_\varphi\rangle\,d\varrho d\varphi+|Z_\varphi|^2\,d\varphi^2 \\[0.6ex] &= d\varrho^2+P(\varrho,\varphi)\,d\varphi^2\,, \end{align} see, for example, → W. Blaschke and K. Leichtweiß 1973, §79. The function \( P\in C^1((0,r]\times[0,2\pi),\mathbb R) \) satisfies \[ P(\varrho,\varphi)\gt 0\quad\mbox{for all}\ (\varrho,\varphi)\in(0,r]\times[0,2\pi) \] as well as \[ \lim_{\varrho\to 0_+}P(\varrho,\varphi)=0,\quad \lim_{\varrho\to 0_+}\frac{\partial}{\partial\varrho}\,\sqrt{P(\varrho,\varphi)}=1 \quad\mbox{for all}\ \varphi\in[0,2\pi). \] We will refer to these relations when we establish energy estimates for geodesic discs, in particular in the context of curvature estimates.
We will usually use special coordinate systems which are compatible with the special problems, but, eventually, we will formulate results also using coordinate-independent vector fields to emphasise their geometric importance. Let us consider the following example: The components \( g_{ij} \) of the first fundamental form \( I(X) \) can be considered as the components of a bi-linear mapping \[ g\colon T_X(w)\times T_X(w)\longrightarrow\mathbb R,\quad X_{u^i},X_{u^j}\mapsto g_{ij}:=g(X_{u^i},X_{u^j}). \] In this way we could define the metric \( g \) without referring to a special parametrization. Namely, let \( \langle\cdot,\cdot\rangle(w) \) be a positive definite quadratic form on \( T_X(w)\times T_X(w), \) then we set \[ g({\mathcal X},{\mathcal Y})(w):=\langle{\mathcal X},{\mathcal Y})(w) \] for two tangential vectors \( {\mathcal X},{\mathcal Y}\in T_X(w). \) The form \( \langle\cdot,\cdot\rangle(w) \) could result from projecting the metric of the embedding space to the tangential space of the surface, the former represented by the Euclidean inner product - then \( g \) equals our first fundamental form \( I(X) \) from above.
2.5.1 Regular parameter transformations
\( \require{boldsymbol} \)
Definition: The \( C^{4+\alpha} \)-regular parameter transformation \[ (\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})\mapsto \psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =(u(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}),v(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})), \quad(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})\in\overset{\boldsymbol{\,\sim}}{B}\subset\mathbb R^2\,, \] is called of regularity class \( {\mathfrak P}, \) shortly: \( \psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})\in{\mathfrak P}, \) if it represents a diffeomorphism between \( \overset{\boldsymbol{\,\sim}}{B} \) and \( B \) with the property \[ \mbox{det}\,D\psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =\mbox{det} \left( \begin{array}{cc} \displaystyle\frac{\partial u}{\partial\overset{\boldsymbol{\sim}}{u}} & \displaystyle\frac{\partial u}{\partial\overset{\boldsymbol{\sim}}{v}} \\[0.6ex] \displaystyle\frac{\partial v}{\partial\overset{\boldsymbol{\sim}}{u}} & \displaystyle\frac{\partial v}{\partial\overset{\boldsymbol{\sim}}{v}} \end{array} \right) \not=0\quad\mbox{in}\ \overset{\boldsymbol{\sim}}{B} \] with the Jacobian matrix \( D\psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}). \) The mapping \[ \overset{\boldsymbol{\,\sim}}{X}(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) :=X(u(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}),v(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})) \] is then called a regular reparametrization of \( X(u,v). \)
Remark: If \( \mbox{det}\,D\psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})\gt 0, \) the parameter transformation \( \psi \) is called orientation-preserving, otherwise orientation-reversing.
For the tangential vectors of the surface vector we compute \[ \overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^m} =\sum_{i=1}^2X_{u^i}\frac{\partial u^i}{\partial{\overset{\boldsymbol{\sim}}{u}^m}} =\sum_{i=1}^2\Lambda_m^i X_{u^i} \quad\mbox{with}\ \Lambda_m^i:=\frac{\partial u^i}{\partial{\overset{\boldsymbol{\sim}}{u}^m}} \] using the chain rule.
2.5.2 Transformation of the first fundamental form
Using the transformation rule for \( \overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^m} \) from the previous paragraph, we compute the coefficients of the first fundamental form of \( \overset{\boldsymbol{\,\sim}}{X} \) \[ \overset{\boldsymbol{\,\sim}}{g}_{mn} =\langle\overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^m},\overset{\boldsymbol{\,\sim}}{X}_{\overset{\boldsymbol{\sim}}{u}^n}\rangle =\sum_{i,j=1}^2\Lambda_m^i\Lambda_n^j\langle X_{u^i},X_{u^j}\rangle =\sum_{i,j=1}^2\Lambda_m^i\Lambda_n^jg_{ij}\,,\quad m,n=1,2. \] It is often advantageous to have this identity in terms of the Jacobian \[ D\psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =\left( \begin{array}{cc} \Lambda_1^1 & \Lambda_2^1 \\[0.6ex] \Lambda_1^2 & \Lambda_2^2 \end{array} \right). \] and its transpose \( D\psi^T. \) Namely, introducing the matrices \[ \boldsymbol{g}:=(g_{ij})_{i,j=1,2}\,,\quad \overset{\boldsymbol{\,\sim}}{\boldsymbol{g}}:=(\overset{\boldsymbol{\,\sim}}{g}_{mn})_{m,n=1,2}\, \] it holds \[ \overset{\boldsymbol{\,\sim}}{\boldsymbol{g}}=D\psi^T\circ\boldsymbol{g}\circ D\Psi\quad\mbox{in}\ \overset{\boldsymbol{\,\sim}}{B}\,. \]
2.5.3 Transformation of the area element
An immediate consequence of the identities from the previous paragraph is
Proposition: Let \( \psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) \) be a positively oriented parameter transformation of class \( {\mathfrak P}. \) Then it holds \[ \widetilde W(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v}) =\mbox{det}\,\psi(\overset{\boldsymbol{\sim}}{u},\overset{\boldsymbol{\sim}}{v})\cdot W(u,v). \]
Proof: We compute \[ \widetilde W =\sqrt{\mbox{det}\,\overset{\boldsymbol{\,\sim}}{\boldsymbol{g}}} =\sqrt{\mbox{det}\,D\psi^T\cdot\mbox{det}\,\boldsymbol{g}\cdot\mbox{det}\,D\psi} =\mbox{det}\,\psi\cdot\sqrt{\boldsymbol{g}} =\mbox{det}\,\psi\cdot W \] since \( \mbox{det}\,\psi\gt 0 \) by assumption. The proof is complete.\( \qquad\Box \)