Total Gaussian Curvature

Abstract

If we know a plane curve $\gamma \colon [a,b]\to \mathbb {R}^2$ near its end points, we know its total curvature $\int _a^b \kappa \,ds$ up to an integer multiple of $2\pi$.

If we know a plane curve $\gamma \colon [a,b]\to \mathbb {R}^2$ near its end points, we know its total curvature $\int _a^b \kappa \,ds$ up to an integer multiple of $2\pi$. This follows from the results in Chap. 3. Here we prove a similar result for surfaces $f\colon M\to \mathbb {R}^3$ in three-space: If we know f near the boundary of M, we know its total Gaussian curvature $\int _M K \,\det$ up to an integer multiple of $2\pi$. However, unlike the situation for plane curves, the integer in question is already completely determined by the topology of M.

1 Curves on Surfaces

Definition 10.1

Let $f\colon M\to \mathbb {R}^3$ be a surface with unit normal field N and $\gamma \colon [a,b]\to M$ a curve in M. Then the pair $(\gamma ,f)$ is called a curve on the surfacef. The space curve

$$\displaystyle \begin{aligned} \tilde{\gamma}=f\circ \gamma\end{aligned}$$

is called the trace of $(\gamma ,f)$. The velocity of $(\gamma ,f)$ is defined as $|\tilde {\gamma }'|$ and accordingly the derivative with respect to arclength of a function $g\colon [a,b]\to \mathbb {R}^k$ is to be interpreted as

$$\displaystyle \begin{aligned} \frac{dg}{ds}:=\frac{g'}{|\tilde{\gamma}'|}.\end{aligned}$$

The unit tangent $\tilde {T}$ of $\tilde {\gamma }$ is called the unit tangent of $(\gamma ,f)$ and the unit normal field

$$\displaystyle \begin{aligned} \tilde{N}=N\circ \gamma\end{aligned}$$

along $\tilde {\gamma }$ is called the surface normal of $(\gamma ,f)$. The unit normal field

$$\displaystyle \begin{aligned} \tilde{B}=\tilde{T}\times \tilde{N}\end{aligned}$$

along $\tilde {\gamma }$ is called the binormal of $(\gamma ,f)$.

If $(\gamma ,f)$ is a curve on the surface f, then $(\tilde {\gamma },\tilde {N})$ defined as above will be a framed curve according to Definition 5.11.

Definition 10.2

If $(\gamma ,f)$ is a curve on the surface f and $\tilde {T},\tilde {N},\tilde {B}$ are defined as in Definition 10.1, then

• (i) The normal curvature of $(\gamma ,f)$ is defined as

  $$\displaystyle \begin{aligned} \kappa_n=\langle \tilde{N}',\tilde{T}\rangle.\end{aligned}$$

• (ii) The geodesic curvature of $(\gamma ,f)$ is defined as

  $$\displaystyle \begin{aligned} \kappa_g=\langle \tilde{B}',\tilde{T}\rangle.\end{aligned}$$

• (iii) The geodesic torsion of $(\gamma ,f)$ is defined as

  $$\displaystyle \begin{aligned} \tau=\langle \tilde{N}',\tilde{B}\rangle.\end{aligned}$$

Traditionally, curves on a surface f for which one of these quantities vanishes are designated by special names (see Fig. 10.1):

Fig. 10.1

An asymptotic line (top), a geodesic (middle) and a curvature line (bottom) on a torus

Definition 10.3

Let $(\gamma ,f)$ be a curve on the surface $f\colon M\to \mathbb {R}^3$. Then

• (i) $(\gamma ,f)$ is called an asymptotic line if its normal curvature $\kappa _n$ vanishes.

• (ii) $(\gamma ,f)$ is called a geodesic if its geodesic curvature $\kappa _g$ vanishes.

• (iii) $(\gamma ,f)$ is called a curvature line if its geodesic torsion $\tau$ vanishes.

Remark 10.4

The geodesic in Fig. 10.1 illustrates nicely that geodesics are locally length minimizing, but globally they are not necessarily the shortest path between two points.

2 Theorem of Gauss and Bonnet

Let $M\subset \mathbb {R}^2$ be a compact domain with smooth boundary. By Definition 6.1 and the arguments surrounding Fig. 8.5, each of the n components of the boundary $\partial M$ can be parametrized by a closed curve $\gamma _j\colon [0,2\pi ]\to M$. Given a surface $f\colon M\to \mathbb {R}^3$, we define the total geodesic curvature of the boundary $\partial M$ by summing up the integrals of the geodesic curvature $\kappa _g$ over the corresponding curves $(\gamma _j,f)$ on the surface f (Definitions 10.1 and 10.2):

$$\displaystyle \begin{aligned} \int_{\partial M}\kappa_g := \sum_{j=1}^n \int_{\gamma_j}\kappa_g\,ds.\end{aligned}$$

Definition 10.5

Let $M\subset \mathbb {R}^2$ be a domain with smooth boundary having k components and n boundary curves. Then

$$\displaystyle \begin{aligned} \chi(M):=2k-n\end{aligned}$$

is called the Euler characteristic of M.

Theorem 10.6 (Gauss-Bonnet Theorem)

Let$M\subset \mathbb {R}^2$be a compact domain with smooth boundary having k connected components. Assume that the boundary$\partial M$has n components. Let$f\colon M\to \mathbb {R}^3$be a surface,$K\colon M\to \mathbb {R}$its Gaussian curvature and$\det$its area form. Then

$$\displaystyle \begin{aligned} \int_M K\det + \int_{\partial M}\kappa_g \,ds = 2\chi(M).\end{aligned}$$

Before we proof the theorem we note that we may always choose a vector field $Z\in \Gamma (TM)$ with $\langle Z,Z\rangle =1$, for example one could take $Z=\frac {1}{|U|}U$. Moreover, we will make use of the following helpful observations.

Lemma 10.7

Let$M\subset \mathbb {R}^2$be a compact domain with smooth boundary,$\gamma \colon [-\pi , \pi ]\to \mathbb {R}^2$be a parametrization of a boundary curve of M and$Z\in \Gamma (TM)$be a vector field with$\langle Z,Z\rangle =1$. Then, there is a smooth function$\alpha \colon \mathbb {R}\to \mathbb {R}$with$\alpha (x+2\pi )=\alpha (x)+2\pi \ell$for some$\ell \in \mathbb {Z}$such that

$$\displaystyle \begin{aligned} T=\cos{}(\alpha)Z\circ\gamma + \sin{}(\alpha)JZ\circ \gamma,\end{aligned}$$

where$T=\frac {1}{\sqrt {\langle \gamma ',\gamma '\rangle }}\gamma '$is the unit tangent vector of gamma.

Proof

For each $x\in [-\pi ,\pi ]$ we can choose an $\alpha (x)\in \mathbb {R}$ such that

$$\displaystyle \begin{aligned} T(x)=\cos{}(\alpha(x))Z\circ\gamma(x) + \sin{}(\alpha(x))JZ\circ\gamma(x).\end{aligned}$$

Due to the smoothness of the boundary curve, locally this choice of $\alpha$ can be made in a smooth fashion. Therefore, $\omega :=\alpha '$ is well defined, so that we can safely define

$$\displaystyle \begin{aligned} \alpha(x)=\alpha(0)+\int_0^x\omega.\end{aligned}$$

In particular,

$$\displaystyle \begin{aligned} \int_{-\pi}^\pi\omega = \alpha(\pi)-\alpha(-\pi)=2\pi\ell\end{aligned}$$

for some $\ell \in \mathbb {Z}$, which is exactly the tangent winding number of the curve

$$\displaystyle \begin{aligned} \begin{pmatrix} \langle\gamma', Z\circ\gamma\rangle \\ \langle\gamma',JZ\circ\gamma\rangle \end{pmatrix}\colon[-\pi,\pi]\to\mathbb{R}^2\end{aligned}$$

which we will denote by $\ell (Z,\langle \cdot ,\cdot \rangle )$. □

Lemma 10.8

$\ell (Z,\langle \cdot ,\cdot \rangle )$is independent of the chosen metric.

Proof

Let $\langle \cdot ,\cdot \rangle ^\sim$ be any other metric. Then for any $t\in [0,1]$ also $\langle \cdot ,\cdot \rangle _t:= (1-t)\langle \cdot ,\cdot \rangle + t\langle \cdot ,\cdot \rangle ^\sim$ is again a Riemannian metric. As $\ell$ is an integer and depends continuously on t we conclude that it is constant and therefore $\ell (Z,\langle \cdot ,\cdot \rangle )=\ell (Z,\langle \cdot ,\cdot \rangle ^\sim )$. □

Theorem 10.9

Let$M\subset \mathbb {R}^2$be a compact domain with smooth boundary having k connected components and n boundary curves. Let$\gamma _1,\ldots ,\gamma _n$be closed parametrizations of the n boundary curves of M and$\ell _1,\ldots ,\ell _n$be the tangent winding numbers of$\gamma _1,\ldots ,\gamma _n$with respect to a unit vector field$Z\in \Gamma (TM)$. Then

$$\displaystyle \begin{aligned} \sum_{j=1}^n\ell(Z)=2\pi\chi(M).\end{aligned}$$

Proof

Without loss of generality let $\langle \cdot ,\cdot \rangle = \langle \cdot ,\cdot \rangle _{\mathbb {R}^2}$ . For $j=1,\ldots ,n$ denote the tangent winding number of $\gamma _j$ by $\ell _j(Z)$. Then either $\ell _j(Z)=1$ (if $\gamma _j$ parametrizes the outer boundary of one of the components of M) or $\ell _j(Z)=-1$. Since there are k components and $n -k$ interior components, we have

$$\displaystyle \begin{aligned} \sum_{j=1}^n\ell_j(Z)=k - (n-k) = 2k-n = \chi(M).\end{aligned}$$

□

Proof of Theorem 10.6

Choose a vector field $Z\in \Gamma (TM)$ with $\langle Z,Z\rangle =1$ and define a 1-form $\eta \in \Omega ^1(M)$ by

$$\displaystyle \begin{aligned} \eta(X)=\langle \nabla_X Z, JZ\rangle.\end{aligned}$$

Think of $\eta (X)$ as the rotation speed of Z in the direction of X. Because of $\langle \nabla _XZ,Z\rangle =0$ (which follows from differentiating $\langle Z,Z\rangle =1$) and and since $Z,JZ$ is a positively oriented basis of $T_pM$ we must have

$$\displaystyle \begin{aligned} \nabla_XZ=\eta(X)JZ.\end{aligned}$$

Using this, (ii) of Theorem 9.2, the Gauss equation (Theorem 9.8) and (i) of Theorem 9.4 we find

$$\displaystyle \begin{aligned} d\eta(U,V)&=d_U\eta(V)-d_V\eta(U) \\&=\langle \nabla_U\nabla_V Z,JZ\rangle +\langle \nabla_V Z, J\nabla_UZ\rangle \\ &\quad - \langle \nabla_V\nabla_U Z,JZ\rangle -\langle \nabla_U Z, J\nabla_VZ\rangle \\&= \langle \nabla_U\nabla_V Z,JZ\rangle - \langle \nabla_V\nabla_U Z,JZ\rangle \\&= -K\det(U,V)\end{aligned} $$

and therefore

$$\displaystyle \begin{aligned} d\eta=-K\det.\end{aligned}$$

In particular, this means that $d\eta$ does not depend on our choice of Z. Therefore Stokes’ theorem implies

$$\displaystyle \begin{aligned} -\int_MK\det = \int_{\partial M}\eta.\end{aligned}$$

For each of the boundary components, the geodesic curvature of $\tilde \gamma _j:= f\circ \gamma _j$ can be expressed as $\kappa _j:=\langle \tilde T_j^{\prime },\tilde N\times \tilde T_j\rangle$. Then, with

$$\displaystyle \begin{aligned} \left(df(Z\circ \gamma)\right)'=-\langle A\gamma', Z\circ\gamma\rangle \tilde N+df(\nabla_{\gamma'}Z\circ\gamma )\end{aligned}$$

we have

$$\displaystyle \begin{aligned} \kappa_j &= \langle\tilde T_j^{\prime}, \tilde N\times \tilde T\rangle\\ &= \langle (df(T_j))', \tilde N\times df(T)\rangle\\ &= \alpha' + \eta(\gamma'). \end{aligned} $$

Putting everything together we obtain

$$\displaystyle \begin{aligned} \int_{\partial M}\kappa_g &= \sum_{j=1}^n\int_{\gamma_j}\kappa_j \\ &= \sum_{j=1}^n\int_{\gamma_j}\left(\alpha' +\eta \right) \\ &= 2\pi\sum_{j=1}^n\ell_j(Z) + \int_{\partial M}\eta \\ &= 2\pi\chi(M) - \int_{M}K\det . \end{aligned} $$

□

It is quite striking that the total amount of Gaussian curvature (in the sense of $\int _M K\,\det$) is completely determined by the geometry of f near the boundary of M (see Fig. 10.2).

Fig. 10.2

Even if we do not know the shape of a rounded cone near its tip (only revealed under a microscope), the integral of the Gaussian curvature can be deduced from the opening angle of the cone

Example 10.10

Suppose M is a disk which is mapped to the top half of a round sphere with radius $r>0$ by f. Then the boundary curve lies on the equator which is known to be a geodesic, i.e. $\kappa _g=0$. Therefore, the Gauss-Bonnet theorem yields

$$\displaystyle \begin{aligned} \int_MK\det = \tfrac{1}{r^2}\mathcal{A}(f(M)) = 2\pi.\end{aligned}$$

Example 10.11

If M is an annulus and f maps it onto a cylinder, then $\chi (M)=0$ and $K=0$, so the Gauß-Bonnet formula yields

$$\displaystyle \begin{aligned} \int_{\partial M}\kappa_g\,ds=0.\end{aligned}$$

3 Parallel Transport on Surfaces

In Sect. 5.1 we studied the normal transport $\mathcal {P}\colon T(a)^\perp \to T(b)^\perp$ of a curve $\gamma \colon [a,b]\to \mathbb {R}^3$ with unit tangent T. A closer look reveals that in order to define $\mathcal {P}$ only the smooth map $T\colon [a,b]\to S^2$ is needed. Therefore, given a surface $f\colon M\to \mathbb {R}^3$ and a smooth map $\gamma \colon [a,b]\to M$, we can we can use the same strategy in order to transport tangent vectors $W\in T_{\gamma (a)}M$ to tangent vectors $\mathcal {P}(W)\in T_{\gamma (b)}M$:

Definition 10.12

Let $f\colon M\to \mathbb {R}^3$ be a surface with unit normal field N and $\gamma \colon [a,b]\to M$ a smooth map. Define $\tilde {N}\colon [a,b]\to S^2$ by

$$\displaystyle \begin{aligned} \tilde{N}:=N\circ \gamma\end{aligned}$$

and for $W\in T_{\gamma (a)}M$ define the parallel transport map$\mathcal {P}_\gamma (W)\in T_{\gamma (b)}M$ in such a way that

$$\displaystyle \begin{aligned} df(\mathcal{P}(W)):= Z(b)\end{aligned}$$

where $Z\colon [a,b]\to \mathbb {R}^3$ solves the initial value problem

$$\displaystyle \begin{aligned} Z(a) &=df(W) \\ Z' &=-\langle Z, \tilde{N}\,\!'\rangle \tilde{N}.\end{aligned} $$

$\tilde {N}$ plays exactly the same role here as T did in Sect. 5.1. Hence, for the same reasons as in Sect. 5.1, we have $\langle Z,\tilde {N}\rangle =0$ and indeed for all $x\in [a,b]$ the vector $Z(x)$ is an element of $df(T_{\gamma (x)}M)$. Furthermore,

$$\displaystyle \begin{aligned} \mathcal{P}_\gamma\colon T_{\gamma(a)}M \to T_{\gamma(a)}M\end{aligned}$$

is an orientation-preserving orthogonal map with respect to the metrics induced by f on $T_{\gamma (a)}M$ and $T_{\gamma (b)}M$.

The derivative $Z'(x)$ is a multiple of $N(\gamma (x))$, so it has no component in $df(T_{\gamma (x)}M)$. In the spirit of the Sect. 9.1 (Levi-Civita connection), where a derivative $\nabla _XY$ of a vector field Y was defined in terms of the tangential component of $d_X(df(Y))$, this means that $\mathcal {P}$ can be viewed as parallel transport along $\gamma$.

Imagine a pendulum swinging at a point of a surface $f\colon M\to \mathbb {R}^3$ subject to gravity pointing away form the unit normal of the surface. Suppose we transport the swinging pendulum along a path $f\circ \gamma$ where $\gamma \colon [a,b]\to M$ is a smooth map and that the plane in which the pendulum swings initially is given as $df(W)$ where $W\in T_{\gamma (a)}M$ is a unit vector with respect to the induced metric. Then Physics tells us that the plane in which the pendulum swings once it arrives at $f(\gamma (b))$ will be given by the unit vector $df(\mathcal {P}(W))$.

In the special case where f parametrizes the surface of the earth and the movement $\gamma$ corresponds to the rotation of the earth, this effect can be experimentally verified and is known under the name of Foucault’s pendulum (see Fig. 10.3).

Fig. 10.3

In 1851, Léon Foucault build a Foucault pendulum to demonstrate the rotation of the earth (left). The phenomenon can be understood with the concept of parallel transport, where a tangent vector is transported along a circle of latitude (right)

As in Sect. 5.1, if we choose unit vectors (with respect to the induced metric) $W_a\in T_{\gamma (a)}M$ and $W_a\in T_{\gamma (b)}M$, we can measure the parallel transport along $\gamma$ by an angle $\mathcal {P}_W\in \mathbb {R}/_{2\pi \mathbb {Z}}$. For closed curves $\gamma$ this angle does not depend on the choice of $W_a$ and $W_b$ as long as we make sure that $W_a=W_b$. In the special case where $\gamma$ parametrizes the boundary $\partial M$ of M, this angle can be expressed in terms of the total Gaussian curvature of f (see Fig. 10.4).

Fig. 10.4

Parallel transport of a tangent vector along a closed curve on a surface with positive Gaussian curvature (left) and along the boundary of a surface with negative Gaussian curvature (right)

Theorem 10.13

Suppose that$f\colon M\to \mathbb {R}^3$is a surface and that M has only a single boundary component parametrized by a curve$\gamma \colon [a,b]\to M$. Then themonodromy angleof$\gamma$satisfies

$$\displaystyle \begin{aligned} \mathcal{M}(\gamma)\equiv\int_M K\,\det \mod 2\pi\mathbb{Z}\end{aligned}$$

where K is the Gaussian curvature of f.

Proof

Let us assume that $\gamma$ has unit speed with respect to the induced metric and therefore $\tilde {T}:=\tilde {\gamma }'$ is the unit tangent field of $\tilde {\gamma }:=f\circ \gamma$. Define $\tilde {N}:=N\circ \gamma$ where N is the unit normal field of f. Let W and Z be defined as in Definition 10.12. Then there is a smooth function $\alpha \colon [a,b]\to \mathbb {R}$ such that

$$\displaystyle \begin{aligned} Z=\cos \alpha \,\tilde{T} + \sin \alpha\, \tilde{N}\times \tilde T.\end{aligned}$$

We denote by $\kappa _g=\langle \tilde {T}',\tilde {N}\times \tilde {T}\rangle$ the binormal curvature of the framed curve $(\tilde \gamma ,\tilde {N})$. Because $Z'$ is normal, we have

$$\displaystyle \begin{aligned} 0 =\langle Z',\tilde{N}\times Z \rangle = \alpha'+\kappa_g.\end{aligned}$$

Finally, by the Gauss-Bonnet Theorem 10.6 we have

$$\displaystyle \begin{aligned} \mathcal{M}(\gamma) \equiv \alpha(b)-\alpha(a)=\int_a^b \alpha' =-\int_a^b \kappa_g \equiv \int_M K\,\det \mod 2\pi\mathbb{Z}.\end{aligned}$$

□