1. Surfaces in Euclidean spaces
1.1.1 Immersions on the unit disc
Let \( n\ge 1 \) be a natural number. We mainly study two-dimensional immersions of disc-type in parametric form with trace in Euclidean space \( \mathbb R^{n+2}. \) More precisely, we consider two-dimensional vector-valued mappings \[ X=X(u,v)=(x^1(u,v),\ldots,x^{n+2}(u,v))\in C^{4+\alpha}(B,\mathbb R^{n+2}),\quad\alpha\in(0,1), \] on the closed unit disc \[ B:=\{(u,v)\in\mathbb R^2\,:\,u^2+v^2\le 1\}\subset\mathbb R^2\,. \] Furthermore, we set \[ \mathring B:=\{(u,v)\in\mathbb R^2\,:\,u^2+v^2\lt 1\} \] for the open unit disc, i.e. \( \mathring B \) denotes the interior of \( B, \) and we write \[ \partial B:=\{(u,v)\in\mathbb R^2\,:\,u^2+v^2=1\} \] for its boundary. Many of our considerations will require simply connected and smoothly bounded domains of definition, which is why we choose \( B \) as parameter domain.
1.1.2 The concept of an immersion
Furthermore, we assume that our mappings \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}), \) \( \alpha\in(0,1), \) are also regular in the following differential geometric sense \[ \mbox{rank}\,DX =\mbox{rank}\left(\begin{array}{cc} x_u^1 & x_v^1 \\ \vdots & \vdots \\ x_u^{n+2} & x_v^{n+2} \end{array}\right)=2\quad\mbox{in}\ B, \] i.e. they represent immersions, where \( DX \) denotes the Jacobian of \( X, \) and the lower indices \( u \) and \( v \) mean the partial derivatives w.r.t. the respective parameters. This condition ensures that at each point \( (u,v)\in B \) there are two linearly independent tangential vectors \( X_u \) and \( X_v \) of the mapping \( X, \) simultaneously its partial derivatives w.r.t. \( u \) or \( v, \) which span the two-dimensional tangential space \[ T_X(w):=\mbox{span}\,\{X_u(w),X_v(w)\}\,,\quad w=(u,v)\in B, \] attached at the point \( w\in B. \) Finally, the regularity condition \( \mbox{rank}\,DX=2 \) in \( B \) ensures that the surface area element \[ W:=\sqrt{\langle X_u,X_u\rangle\langle X_v,X_v\rangle-\langle X_u,X_v\rangle^2} \] of \( X \) is positive: \( W\gt 0 \) in \( B, \) where \[ \langle X,Y\rangle:=\sum_{i=1}^{n+2}x^iy^i \] denotes the Euclidean inner product for two vectors \( X \) and \( Y. \)
Definition: We call \( X\in C^{4+\alpha}(B,\mathbb R^{n+2}) \) a regular parametrization or regular surface or, for short, immersion if it satisfies the differential geometric regularity condition \( \mbox{rank}\,DX=2 \) in \( B. \)
1.1.3 Normal space and normal bundle
For a regular surface we are lead to the decomposition \[ \mathbb R^{n+2}=T_X(w)\cup N_X(w),\quad T_X(w)\perp N_X(w), \] with the \( n \)-dimensional normal space \[ N_X(w):=\{Z\in\mathbb R^{n+2}\,:\,\langle Z,X_u(w)\rangle=\langle Z,X_v(w)\rangle=0\} \] attached at \( w\in B. \) The normal space of a two-dimensional surface in \( \mathbb R^3 \) consists only of the linear span of one normal vector. In our general situation, the normal space at each point has \( n \) dimensions, and, as we will see, the normal bundle \[ \dot{\bigcup_{w\in B}}\{w\}\times N_X(w) \] possesses an own geometry with an an own nontrivial curvature tensor.
A surface graph in \( \mathbb R^4 \) has the form \[ X(u,v)=(u,v,\varphi(u,v),\psi(u,v)),\quad(u,v)\in B, \] with functions \( \varphi,\psi\in C^4(B,\mathbb R). \) Its tangential vectors are \[ X_u=(1,0,\varphi_u,\psi_u),\quad X_v=(0,1,\varphi_v,\psi_v), \] and we conclude \begin{align} \langle X_u,X_v\rangle &= 1+\varphi_u^2+\psi_u^2\,, \\[0.6ex] \langle X_v,X_v\rangle &= 1+\varphi_v^2+\psi_v^2\,, \\[0.6ex] \langle X_u,X_v\rangle &= \varphi_u\varphi_v+\psi_u\psi_v\,, \end{align} and, thus, \[ W=\sqrt{1+|\nabla\varphi|^2+|\nabla\psi|^2+(\varphi_u\psi_v-\varphi_v\psi_u)^2} \] for the area element \( W. \) The general situation for graphs in \( \mathbb R^{n+2} \) is analogous.
1.2.2 The complexified Neile parabola
The classical Neile's parabola is the curve \[ c(t)=(t^2,t^3),\quad t\in\mathbb R, \] with a cusp at the point \( t=0. \) Its complexified version \[ X(w)=(w^2,w^3),\quad w\in B, \] now interpreting \( w=u+iv \) as a complex number, or in real terms \[ X(u,v)=(u^2-v^2,2uv,u^3-3uv^2,3u^2v-v^3),\quad(u,v)\in B, \] is not regular at \( (u,v)=(0,0) \) since the tangential vectors \[ X_u=(2u,2v,3u^2-3v^2,6uv),\quad X_v=(-2v,2u,-6uv,3u^2-3v^2) \] degenerate into a point at \( (u,v)=(0,0). \)
1.2.3 The Henneberg minimal surface
This surface has the possible parametrization \begin{align} x^1(u,v) & = 2\sinh u\cos v-\frac{2}{3}\,\sinh(3u)\cos(3v), \\[1ex] x^2(u,v) & = 2\sinh u\sin v-\frac{2}{3}\,\sinh(3u)\sin(3v), \\[1ex] x^3(u,v) & = 2\cosh(2u)\cos(2v). \end{align} We calculate \begin{align} x_u^1 & = 2\cosh u\cos v-2\cosh(3u)\cos(3v), \\[0.6ex] x_u^2 & = 2\cosh u\sin v-2\cosh(3u)\sin(3v), \\[0.6ex] x_u^3 & = 4\sinh(2u)\cos(2v) \end{align} as well as \begin{align} x_v^1 &= -2\sinh u\sin v+2\sinh(3u)\sin(3v), \\[0.6ex] x_v^2 &= 2\sinh u\cos v-2\sinh(3u)\cos(3v), \\[0.6ex] x_v^3 &= -4\cosh(2u)\sin(2v) \end{align} and, therefore, \[ X_u(0,0)=(0,0,0),\quad X_v(0,0)=(0,0,0). \] Thus, our parametrization is not regular at the origin, and in fact, the Henneberg surface as a surface in space has a singular point at \( X(0,0). \) An image of the surface can be found → here.
Our interest is primarily focused on the analysis of two-dimensional immersions together with suitable vector systems spanning their normal spaces.
Definition: Let \( X \) be a regular parametrization. A system \( {\mathfrak N}=(N_1,\ldots,N_n) \) consisting of \( n \) mappings \( N_\sigma\in C^{3+\alpha}(B,\mathbb R^{n+2}), \) \( \alpha\in(0,1), \) which satisfy \[ \langle N_\sigma,N_\vartheta\rangle =\delta_{\sigma\vartheta} :=\left\{\begin{array}{cl} 1 & \quad\mbox{if}\ \sigma=\vartheta \\ 0 & \quad\mbox{if}\ \sigma\not=\vartheta \end{array}\right., \quad\sigma,\vartheta=1,\ldots,n, \] where \( \delta_{\sigma\vartheta} \) denotes the Kronecker symbol, as well as \[ N_X(w)=\mbox{span}\,\{N_1(w),\ldots,N_n(w)\}\,, \] is called an orthonormal normal frame or ONF of the surface \( X. \)
There always exist ONFs for regular surfaces \( X \) defined the parameter domain \( B \) due to the contractibility of \( B, \) see, for example, section 2.1 in → S. Parr 2021, section 2.1.
Example: The unit normal vector \[ N(u,v):=\frac{X_u(u,v)\times X_v(u,v)}{|X_u(u,v)\times X_v(u,v)|} \] of an immersion \( X\colon B\to\mathbb R^3 \) is a trivial example of an ONF since it consists of only one vector.
The following paragraphs collect further examples of surfaces for which we can specify explicitely ONFs.
1.3.2 ONFs for holomorphic surfaces
A holomorphic surface is an immersion in \( \mathbb R^4\cong\mathbb C^2 \) of the form \[ X(u,v)=(\Phi(u,v),\Psi(u,v)),\quad(u,v)\in B, \] with complex-valued and holomorphic functions \( \Phi=\varphi_1+i\varphi_2 \) and \( \Psi=\psi_1+i\psi_2 \) satisfying in \( B \) the Cauchy-Riemann differential equations \[ \begin{array}{l} \varphi_{1,u}=\varphi_{2,v}\,,\quad \varphi_{1,v}=-\varphi_{2,u}\,, \\[0.6ex] \psi_{1,u}=\psi_{2,v}\,,\quad \psi_{1,v}=-\psi_{2,u}\,. \end{array} \] The tangential vectors are \[ X_u=(\varphi_{1,u},\varphi_{2,u},\psi_{1,u},\psi_{2,u}),\quad X_v=(\varphi_{1,v},\varphi_{2,v},\psi_{1,v},\psi_{2,v}), \] are supposed to be linearly independent, in particular \( X_u\not=0 \) and \( X_v\not=0, \) and we confirm \begin{align} |X_u|^2 &= \varphi_{1,u}^2+\varphi_{2,u}^2+\psi_{1,u}^2+\psi_{2,u}^2 \\[0.6ex] &= \varphi_{2,v}^2+\varphi_{1,v}^2+\psi_{2,v}^2+\psi_{1,v}^2 \\[0.6ex] &= |X_v|^2\,, \\[1ex] \langle X_u,X_v\rangle &= \varphi_{1,u}\varphi_{1,v}+\varphi_{2,u}\varphi_{2,v}+\psi_{1,u}\psi_{1,v}+\psi_{2,u}\psi_{2,v}=0\quad\mbox{in}\ B \end{align} considering the Cauchy-Riemann equations. Furthermore, we define the unit normal vecors \[ N_1:=\frac{(-\psi_{1,u},\psi_{2,u},\varphi_{1,u},-\varphi_{2,u})}{\sqrt{|\Phi_u|^2+|\Psi_u|^2}}\,,\quad N_2:=\frac{(\psi_{1,v},-\psi_{2,v},-\varphi_{1,v},\varphi_{2,v})}{\sqrt{|\Phi_v|^2+|\Phi_v|^2}} \] which satisfy \begin{align} \langle X_u,N_1\rangle=\langle X_v,N_1\rangle &= 0, \\[0.6ex] \langle X_u,N_2\rangle=\langle X_v,N_1\rangle &= 0, \\[0.6ex] \langle N_1,N_2\rangle &= 0 \quad\mbox{in}\ B, \end{align} and, therefore, \( (N_1,N_2) \) represents an ONF of the surface.
Example: Our standard example of a holomorphic surface is \[ X(w)=(w^m,w^n),\quad m,n\in\mathbb N. \] Note that \( X \) is not regular at \( w=0 \) if \( m,n\gt 1. \)
A surface graph has the form \[ X(x,y)=(x,y,\zeta_1(x,y),\ldots,\zeta_n(x,y)),\quad(x,y)\in B, \] with functions \( \zeta_\sigma\in C^{4+\alpha}(B,\mathbb R) \) for \( \sigma=1,\ldots,n. \) Surface graphs are always regular since their tangential vectors \[ X_x=(1,0,\zeta_{1,x},\ldots,\zeta_{n,x}),\quad X_y=(0,1,\zeta_{1,y},\ldots,\zeta_{n,y}) \] do not vanish and are linearly independent. We define the Euler unit normal vectors \begin{align} \overline N_1 &:= \frac{1}{\sqrt{1+|\nabla\zeta_1|^2}}\,(-\zeta_{1,x},-\zeta_{1,y},1,0,0,\ldots,0), \\[1ex] \overline N_2 &:= \frac{1}{\sqrt{1+|\nabla\zeta_2|^2}}\,(-\zeta_{2,x},-\zeta_{2,y},0,1,0,\ldots,0) \\[1ex] &\qquad\qquad\vdots \\[1ex] \overline N_n &:= \frac{1}{\sqrt{1+|\nabla\zeta_n|^2}}\,(-\zeta_{n,x},-\zeta_{n,y},0,0,0,\ldots,1) \end{align} for \( X \) which satify \[ \langle X_x,\overline N_\sigma\rangle=\langle X_y,\overline N_\sigma\rangle=0\quad\mbox{in}\ B \] for all \( \sigma=1,\ldots,n. \) But note that, in general, these \( n \) vectors are not orthonormal. Rather, we must apply the Gram-Schmidt process to obtain an ONF \( (N_1,\ldots,N_n) \) from the \( \overline N_\sigma. \)
Example: Let \( \Phi=\varphi_1+i\varphi_2 \) be holomorphic on \( B, \) then \[ X(u,v)=(u,v,\varphi_1(u,v),\varphi_2(u,v)),\quad(u,v)\in B, \] is a holomorphic graph with Euler unit normal vectors \begin{align} N_1 &= \frac{1}{\sqrt{1+|\nabla\varphi_1|^2}}\,(-\varphi_{1,u},-\varphi_{1,v},1,0), \\[1ex] N_2 &= \frac{1}{\sqrt{1+|\nabla\varphi_2|^2}}\,(-\varphi_{2,u},-\varphi_{2,v},0,1). \end{align} In view of the Cauchy-Riemann equations it holds \( \varphi_{1,u}\varphi_{2,u}+\varphi_{1,v}\varphi_{2,v}=0 \) and, therefore, \[ \langle N_1,N_2\rangle=0\quad\mbox{in}\ B. \] Thus, \( (N_1,N_2) \) represents an ONF for the holomorphic graph \( X. \)
1.3.4 ONFs for spherical surfaces
More precisely, here we mean surfaces \( X\colon B\to\mathbb R^4 \) with the characterizing property \[ |X(u,v)|^2=1\quad\mbox{for all}\ (u,v)\in B. \] Differentiation of this relation yields \[ \langle X,X_u\rangle=\langle X,X_v\rangle=0\quad\mbox{in}\ B \] and, thus, \( N_1:=X \) itself represents an unit normal vector of \( X. \) A second unit normal vector \( N_2 \) follows after completion of \( \{X_u,X_v,N_1\} \) to a right-handed frame \( \{X_u,X_v,N_1,N_2\} \) spanning \( \mathbb R^4, \) and this works as follows: We set \[ N_2=(n_2^1,n_2^2,n_2^3,n_2^4):=-\,\frac{X_u\times X_v\times N_1}{|X_u\times X_v\times N_1|} \] for the second unit normal vector \( N_2. \) By definition, the vector \( X_u\times X_v\times N_1 \) has the components \[ \begin{array}{l} \displaystyle (X_u\times X_v\times N_1)^1 :=+\,\mbox{det} \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ x_u^1 & x_u^2 & x_u^3 & x_u^4 \\ x_v^1 & x_v^2 & x_v^3 & x_v^4 \\ n_1^1 & n_1^2 & n_1^3 & n_1^4 \end{array} \right), \\[2ex] (X_u\times X_v\times N_1)^2 :=-\,\mbox{det} \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ x_u^1 & x_u^2 & x_u^3 & x_u^4 \\ x_v^1 & x_v^2 & x_v^3 & x_v^4 \\ n_1^1 & n_1^2 & n_1^3 & n_1^4 \end{array} \right), \\[2ex] (X_u\times X_v\times N_1)^3 :=+\,\mbox{det} \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ x_u^1 & x_u^2 & x_u^3 & x_u^4 \\ x_v^1 & x_v^2 & x_v^3 & x_v^4 \\ n_1^1 & n_1^2 & n_1^3 & n_1^4 \end{array} \right), \\[2ex] (X_u\times X_v\times N_1)^4 :=-\,\mbox{det} \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ x_u^1 & x_u^2 & x_u^3 & x_u^4 \\ x_v^1 & x_v^2 & x_v^3 & x_v^4 \\ n_1^1 & n_1^2 & n_1^3 & n_1^4 \end{array} \right), \end{array} \] see for example the textbook → H. Grauert, H.-Ch. Grunau 1999, page 202 ff. for this generalization of the usual vector product, and also → S. Parr 2021, page 26 f. The minus sign comes from the even space dimension \( n+2=4. \) Finally, we arrive at an ONF \( (N_1,N_2) \) of the spherical surface \( X. \)
1.3.5 ONFs for the Clifford torus
The Clifford torus, is the product \( \frac{1}{\sqrt{2}}\,S^1\times S^1, \) and a suitable part of it can be parameterized as \[ X(u,v)=\frac{1}{\sqrt{2}}\,(\cos u,\sin u,\cos v,\sin v),\quad(u,v)\in B, \] with tangential vectors \begin{align} X_u & =\frac{1}{\sqrt{2}}\,(-\sin u,\cos u,0,0), \\[1ex] X_v & =\frac{1}{\sqrt{2}}\,(0,0,-\sin v,\cos v). \end{align} The mapping \( X \) obviously represents a spherical surface. We assign a first unit normal vector \[ N_1:=\frac{1}{\sqrt{2}}\,(\cos u,\sin u,\cos v,\sin v). \] Secondly, we compute the components of \( X_u\times X_v\times N_1, \) \begin{align} (X_u\times X_v\times N_1)^1 &= -\,\frac{\cos u}{2\sqrt{2}}\,, \\[1ex] (X_u\times X_v\times N_1)^2 &= -\,\frac{\sin u}{2\sqrt{2}}\,, \\[1ex] (X_u\times X_v\times N_1)^3 &= -\,\frac{\cos v}{2\sqrt{2}}\,, \\[1ex] (X_u\times X_v\times N_1)^4 &= -\,\frac{\sin v}{2\sqrt{2}}\,, \end{align} and infer \[ |X_u\times X_v\times N_1|^2=\frac{1}{8}\,(\cos^2u+\sin^2u+\cos^2v+\sin^2v)=\frac{1}{4}\,. \] Now we have a second unit normal vector \[ N_2:=\frac{1}{\sqrt{2}}\,(\cos u,\sin u,-\cos v,-\sin v) \] and arrive at an ONF \( (N_1,N_2). \)
1.3.6 ONFs for surfaces of Killing
The Clifford torus can be embedded into a two-parameter family of surfaces \[ X(u,v):=\frac{1}{\sqrt{2(\alpha^2+\beta^2)}} \left( \begin{array}{c} \alpha\cos u+\beta\cos v \\ \alpha\sin u-\beta\sin v \\ \alpha\cos v-\beta\cos u \\ \alpha\sin v+\beta\sin u \end{array} \right),\quad \alpha,\beta\in\mathbb R,\ \alpha^2+\beta^2\gt 0, \] in the sphere, i.e. \[ |X(u,v)|=1\quad\mbox{for all}\ (u,v)\in B. \] For \( \alpha=1, \) \( \beta=0 \) or \( \alpha=0, \) \( \beta=1 \) we get the Clifford torus which was considered by Killing → W. Killing 1885, page 241, in the general form \[ x_1=a\cos\frac{u}{a}\,,\quad x_2=a\sin\frac{u}{a}\,,\quad x_3=b\cos\frac{v}{b}\,,\quad x_4=b\sin\frac{v}{b} \] with parameters \( a,b\in\mathbb R. \) For \( \alpha=1, \) \( \beta=1 \) we have \[ X(u,v)=\frac{1}{2}\,(\cos u+\cos v,\sin u-\sin v,-\cos u+\cos v,\sin u+\sin v) \] which was considered in → M. Pinl 1950 and attributed to Killing. All these surfaces can be equipped with ONFs as explained in the foregoing paragraph.
1.3.7 ONFs for set products of curves
The Clifford torus parametrization \[ \frac{1}{\sqrt{2}}\,(\cos u,\sin u,\cos v,\sin v), \] or the parametrization \[ \left(a\cos\frac{u}{a}\,,a\sin\frac{u}{a}\,,b\cos\frac{v}{b}\,,b\sin\frac{v}{b}\right) \] from Killing → W. Killing 1885, page 241, are examples of surfaces which arise as set-theoretic product of two plane curves \[ c(u):=(x^1(u),x^2(u),0,0),\quad \widetilde c(v):=(0,0,x^3(v),x^4(v)). \] If \( c,\widetilde c\in C^{3+\alpha}(I,\mathbb R^2), \) with \( I\subset\mathbb R \) being any suitable interval, are regular in the sense that \[ |\dot c(u)|\not=0\quad\mbox{and}\quad|\dot{\widetilde c}(v)|\not=0, \] where the dot means the derivatives w.r.t. the parameters, the vectors \[ N_1(u,v):=\frac{1}{|\dot c(u)|}\,(-\dot x^2(u),\dot x^1(u),0,0),\quad N_2(u,v):=\frac{1}{|\dot{\widetilde c}(v)|}\,(0,0,-\dot x^4(v),\dot x^3(v)) \] are linearly independent and orthogonal to the tangential vectors \[ X_u(u,v)=(\dot x^1(u),\dot x^2(u),0,0),\quad X_v(u,v)=(0,0,\dot x^3(v),x^4(v)). \] Considering that \( (X_u,X_v,N_2,N_1) \) forms a positively oriented \( 4 \)-frame spanning \( \mathbb R^4 \) in the sense that \[ \mbox{det}\,(X_u,X_v,N_2,N_1)\gt 0, \] For more details we refer to → S. Parr 2021, page 32. the desired ONF is \( (N_2,N_1). \)
1.3.8 ONFs for generalized rotational surfaces
With real numbers \( a,b\in\mathbb R \) and functions \( f,g\in C^{3+\alpha}(\mathbb R,\mathbb R) \) satisfying \[ a^2f(u)^2+b^2g(u)^2\gt 0,\quad f'(u)^2+g'(u)^2\gt 0 \quad\mbox{for all}\ (u,v)\in B \] we consider immersions of the form \[ X(u,v)=(f(u)\cos av,f(u)\sin av,g(u)\cos bv,g(u)\sin bv), \] see → G. Ganchev, V. Milousheva 2008, → M. Stroot 2010, and → S. Parr 2021, section 2.2. The tangential vectors \begin{align} X_u &= (f'\cos av,f'\sin av,g'\cos bv,g'\sin bv), \\[0.6ex] X_v &= (-af\sin av,af\cos av,-bg\sin bv,bg\cos bv) \end{align} are linearly independent and perpendicular to each other since \begin{align} & |X_u|^2=f'^2+g'^2\gt 0,\quad |X_v|^2=a^2f^2+b^2g^2\gt 0, \\[0.6ex] & \langle X_u,X_v\rangle=0\qquad\mbox{in}\ B \end{align} Two unit normal vectors are simple \begin{align} N_1 &= \frac{1}{\sqrt{f'^2+g'^2}}\,(g'\cos av,g'\sin av,-f'\cos bv,-f'\sin bv), \\[1ex] N_2 &= \frac{1}{\sqrt{a^2f^2+b^2g^2}}\,(-bg\sin av,bg\cos av,af\sin bv,-af\cos bv) \end{align} so that \( (N_1,N_2) \) forms an ONF for the surface.