Abstract
From this chapter on we will focus attention on surfaces \(f\colon M \to \mathbb {R}^3\). The most fundamental tool for analysing such a surface is its unit normal field \(N\colon M\to S^2\) which is a map to the unit sphere \(S^2\subset \mathbb {R}^3\).
You have full access to this open access chapter, Download chapter PDF
From this chapter on we will focus attention on surfaces \(f\colon M \to \mathbb {R}^3\). The most fundamental tool for analysing such a surface is its unit normal field \(N\colon M\to S^2\) which is a map to the unit sphere \(S^2\subset \mathbb {R}^3\). The derivative of N reveals information about the curvature of f. In particular, the area covered by N on \(S^2\) provides us with a geometric interpretation of the so-called Gaussian curvature of f.
1 Unit Normal of a Surface in \(\mathbb {R}^3\)
Most of the material in the Chaps. 6 and 7 was concerned with the intrinsic geometry of Riemannian domains or with surfaces in \(\mathbb {R}^n\). From now on we will focus on surfaces \(f\colon M\to \mathbb {R}^3\).
Definition 8.1
Let \(M\subset \mathbb {R}^2\) be a domain with smooth boundary and \(f\colon M\to \mathbb {R}^3\) a surface. Then there is a unique smooth map \(N\colon M\to \mathbb {R}^3\) with \(\langle N,N \rangle =1\) such that
-
(i)
For all \(p\in M\) and all \(X\in T_pM\) we have
$$\displaystyle \begin{aligned} \langle N(p),df(X)\rangle =0.\end{aligned}$$ -
(ii)
For all \(p\in M\) and every positively oriented basis \(X,Y\) of \(T_pM\) we have
$$\displaystyle \begin{aligned} \det(N(p),df(X),df(Y))>0.\end{aligned}$$
N is called the unit normal of f (see Fig. 8.1).
In terms of the coordinate vector fields U and V we can express N as
Theorem 8.2
For all\(p\in M\)and all\(X,Y\in T_pM\)we have
For the area of f we get
Similar as for a surface \(f\colon M\to \mathbb {R}^3\), we can consider the derivative dN of the unit normal \(N\colon M\to \mathbb {R}^3\). In the case of plane curves the derivative of the normal N gave us the curvature \(\kappa \) via the equation
In order to find the analogous equation for surfaces, let us consider a vector field \(X\in \Gamma (TM)\) and take the derivative in the direction of X of the equation \(1=\langle N,N\rangle \):
This means that for all \(X\in T_pM\) the vector \(dN(X)\) lies in the image of the restriction of df to \(T_pM\). Therefore, there is a vector \(Y\in T_pM\) such that \(dN(X)=df(Y)\). Obviously, the dependence of Y on X is linear, so there is a linear map \(A_p\colon T_pM\to T_pM\) such that for all \(X\in T_pM\) we have
We leave it to the reader to check that A is a smooth endomorphism field on M.
Definition 8.3
The smooth endomorphism field A is called the shape operator of f.
Theorem 8.4
The shape operator A is a self-adjoint endomorphism field with respect to the induced metric, i.e. for all\(X,Y\in \Gamma (TM)\)we have
Proof
Since at each point \(p\in M\) the two vectors \(U(p),V(p)\) form a basis of \(T_pM\), it is sufficient to prove the theorem in the special case \(X=U, Y=V\). Using the fact that
we obtain
where we used that the partial derivatives commute, i.e. \(f_{uv}=f_{vu}\). □
2 Curvature of a Surface
The shape operator A of a surface \(f\colon M\to \mathbb {R}^3\) captures all the information about how the surface is curved. In fact it measures deviation from being planar:
Theorem 8.5
Let\(M\subset \mathbb {R}^2\)be a connected compact domain with smooth boundary and\(f\colon M\to \mathbb {R}^3\)a surface with shape operator A. Then A vanishes identically if and only if there is a plane\(E\subset \mathbb {R}^3\)with\(f(M)\subset E\).
Proof
If \(f(M)\subset E\) with
for some unit vector \(\hat {N}\in \mathbb {R}^3\) and \(c\in \mathbb {R}\), then
for all \(X\in TM\), so the unit normal of f satisfies \(N(p)=\pm \hat {N}\) for all \(p\in M\). In particular, \(dN=0\) and therefore \(A=0\).
Conversely, by the connectedness of M, \(A=0\) implies that N is constant, i.e. \(N(p)=\hat {N}\) for some \(\hat {N}\in \mathbb {R}^3\) and all \(p\in M\). Then \(d\langle \hat {N},f\rangle =0\) and (by the connectedness of M) there is \(c\in \mathbb {R}\) such that \(\langle \hat {N},f(p)\rangle = c\) for all \(p\in M\). □
At a given point, a surface can be curved by a different amount in different directions. We call a vector \(X\in TM\) a direction if \(\langle X,X\rangle =1\).
Definition 8.6
For a direction \(X\in TM\) we define the directional curvature\(\kappa (X)\) of f in the direction of X as
If \(X_1,X_2\) is an orthonormal basis of \(T_pM\) then we can parametrize all unit vectors in \(T_pM\) as
Figure 8.2 contains a plot of the function \(\theta \mapsto \kappa (X(\theta ))\).
By Theorem 8.4, for all \(p\in M\) the linear map
is self-adjoint, so there is an orthonormal basis \(X_1,X_2\) in \(T_pM\) such that \(X_1\) and \(X_2\) are eigenvectors of \(A_p\):
If we assume \(\kappa _1\geq \kappa _2\) the eigenvalue functions \(\kappa _1,\kappa _2\colon M \to \mathbb {R}\) are well-defined and continuous. They arise from solving the characteristic equation of \(A_p\), in which a square root is involved. This means that in general (if there are points where \(\kappa _1(p)\) and \(\kappa _2(p)\) coincide) they are not smooth functions.
Definition 8.7
For \(p\in M\) the numbers \(\kappa _1(p)\) and \(\kappa _2(p)\) are called the principal curvatures of f at p. A vector \(X\in T_pM\) with \(\langle X,X\rangle =1\) is called a principal direction corresponding to the principal curvature \(\kappa _j\) if
If we parametrize directions \(X(\theta )\) at p based on principal directions \(X_1,X_2\) as above we obtain
Definition 8.8
The mean value
is called the mean curvature of f at the point p.
We have
so the function \(H\colon M\to \mathbb {R}\) is smooth.
Definition 8.9
The smooth function
is called the Gaussian curvature of f.
If \(K(p)>0\) then the directional curvatures at p are either all positive or all negative. In the first case, the surface looks convex when viewed from “outside” (when we think of N as pointing “outward”). Otherwise it looks concave. Figure 8.3 shows surfaces whose Gaussian curvature is positive everywhere on M.
If \(K(p)<0\) Then the surface bends towards \(N(p)\) is some directions and away from N in other directions. Figure 8.4 shows surfaces whose Gaussian curvature is negative everywhere on M.
Points where the principal curvatures coincide (and therefore all directions are principal directions) are special and we give them a name:
Definition 8.10
A point \(p\in M\) is called an umbilic point of the surface f if at p the surface has the same curvature in all directions, i.e. for all directions \(X\in T_pM\) we have
The most interesting theorems in Differential Geometry lead from local assumptions (curvature properties at each given point) to conclusions about global shape. Here is our first theorem of this kind in the context of surfaces:
Definition 8.11
A subset \(S\subset \mathbb {R}^3\) of the form
with \(\mathbf {m}\in \mathbb {R}^3\) and \(r>0\) is called a round sphere.
Theorem 8.12 (Umbilic Point Theorem)
Let\(M\subset \mathbb {R}^2\)be a connected compact domain with smooth boundary and\(f\colon M\to \mathbb {R}^3\)a surface. Then the following are equivalent:
-
(i)
All points\(p\in M\)are umbilic points.
-
(ii)
Either\(f(M)\subset E\)for some plane\(E\subset \mathbb {R}^3\)or\(f(M)\subset S\)for some round sphere
$$\displaystyle \begin{aligned} S=\left\{\mathbf{p}\in \mathbb{R}^3\,\,|\,\, \langle \mathbf{p}-\mathbf{m},\mathbf{p}-\mathbf{m}\rangle=r^2\right\}.\end{aligned}$$with center\(\mathbf {m}\)and radius\(r>0\).
Proof
If \(f(M)\) is contained in a plane, we already know that \(A=0\) and therefore all points are umbilic points. If \(f(M)\) is contained in a round sphere, then there is a point \(\mathbf {m}\in \mathbb {R}^3\) and a radius \(r>0\) such that
Clearly then, \(f-\mathbf {m}\neq 0\) for all \(p\in M\). Differentiating the above equation reveals that for all \(p\in M\) and all \(X\in T_pM\) we have
Therefore, at each \(p\in M\) the unit normal of f must be given by
By the connectedness of M this implies
and therefore all points are umbilic points:
Conversely, assume that all points \(p\in M\) are umbilic points of f. Then
and therefore \(H_vf_u-H_uf_v=0.\) By the connectedness of M, this means that H is constant. In the case \(H=0\) we have \(A=0\) and by Theorem 8.5 we know that \(f(M)\) is contained in a plane. Otherwise, there is a constant \(r>0\) such that
The function
then satifies \(d\mathbf {m}=0\) and, by the connectedness of M, must be constant. This means that \(f(M)\) lies on a sphere around \(\mathbf {m}\) with radius r. □
3 Area of Maps Into the Plane or the Sphere
Recall the second formula from Theorem 8.2: The area form \(\det \) of a surface \(f\colon M\to \mathbb {R}^3\) with unit normal N is given on \(X,Y\in T_pM\) by
There are situations where we know what N should be, even if f is not a surface but just a smooth map whose derivative \(d_pf\colon T_pM\to \mathbb {R}^3\) might fail to have a two-dimensional image for some \(p\in M\): Define the Euclidean plane\(E^2\) as the subset of \(\mathbb {R}^3\) where the third component is zero. Then at any point \(\mathbf {p}\in E^2\) we consider the third basis vector \({\mathbf {e}}_3\) as the unit normal vector of \(E^2\) at \(\mathbf {p}\). Define the unit two-sphere \(S^2\) as the set of all \(\mathbf {p}\in \mathbb {R}^3\) with \(|\mathbf {p}|=1\). Then at any point \(\mathbf {p}\in S^2\) we consider \(\mathbf {p}\) itself as the unit normal vector of \(S^2\) at \(\mathbf {p}\).
Definition 8.13
Let \(M\subset \mathbb {R}^2\) be a compact domain with smooth boundary and g a smooth map defined on M with values in either \(E^2\) or \(S^2\). Then we define the covered area form\(\sigma _g\in \Omega ^2(M)\) on \(X,Y\in T_pM\) as follows:
-
(i)
For a smooth map \(g\colon M\to E^2\) we define
$$\displaystyle \begin{aligned} \sigma_g(X,Y)=\det\left({\mathbf{e}}_3,dg(X),dg(Y)\right).\end{aligned}$$ -
(ii)
For a smooth map \(g\colon M\to S^2\) we define
$$\displaystyle \begin{aligned} \sigma_g(X,Y)=\det\left(g(p),dg(X),dg(Y)\right).\end{aligned}$$
If we identify \(E^2\) with \(\mathbb {R}^2\) in the obvious way and use the standard determinant \(\det \) on \(\mathbb {R}^2\), the first part of the above definition becomes:
Definition 8.14
Let \(M\subset \mathbb {R}^2\) be a compact domain with smooth boundary and \(g\colon M\to \mathbb {R}^2\) a smooth map. Then we define the covered area form\(\sigma _g\in \Omega ^2(M)\) on \(X,Y\in T_pM\) as
Definition 8.15
Let \(M\subset \mathbb {R}^2\) be a compact domain with smooth boundary and g a smooth map defined on M with values in either \(E^2\), \(S^2\) or \(\mathbb {R}^2\). Then we define the area covered by a mapg as
Given a smooth map \(g\colon M\to \mathbb {R}^2\) from the unit disk M into \(\mathbb {R}^2\), we obtain a loop \(\gamma \colon [0,2\pi ]\to \mathbb {R}^2\) defined by
Conversely, every loop \(\gamma \colon [0,2\pi ]\to \mathbb {R}^2\) arises in this way:
Theorem 8.16
Let\(\gamma \colon \mathbb {R}\to \mathbb {R}^2\)be a loop and\(M\subset \mathbb {R}^2\)the unit disk. Then there is a smooth map\(g\colon M\to \mathbb {R}^2\)such that the Fig.8.5becomes a commutative diagram, i.e.\(\gamma =g\circ s\).
Proof
Let \(\varphi \colon [0,1]\to \mathbb {R}\) be a smooth function such that
Then we can define \(g\colon M\to \mathbb {R}^2\) as the unique map such that for all \(r\in [0,1]\) and all \(t\in \mathbb {R}\) we have
□
Theorem 8.17
Let\(\gamma \colon [0,2\pi ]\to \mathbb {R}^2\)be a loop,\(M\subset \mathbb {R}^2\)the unit disk and\(g\colon M\to \mathbb {R}^2\)any smooth map such that Fig.8.5is a commutative diagram. Then the area covered by g equals the sector area of\(\gamma \).
Proof
Define a 1-form \(\omega \in \Omega ^1(M)\) by setting for \(X\in T_pM\)
Then
Our claim now follows from Stokes Theorem. With
we obtain
□
By Definition 8.13, we obtain a similar interpretation for the area covered by a map \(g\colon M\to S^2\). For us, the most important case is \(g=N\) where N is the unit normal of a surface \(f\colon M\to \mathbb {R}^3\) (see Fig. 8.6):
Theorem 8.18
Let\(f\colon M\to \mathbb {R}^3\)be a surface with unit normal N and Gaussian curvature K. Then the covered area form of N is
Proof
For vector fields \(X,Y\in \Gamma (TM)\) we have
□
Author information
Authors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2024 The Author(s)
About this chapter
Cite this chapter
Pinkall, U., Gross, O. (2024). Curvature. In: Differential Geometry. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-39838-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-39838-4_8
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-39837-7
Online ISBN: 978-3-031-39838-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)