[Tu Differential Geometry] 3. Surface in Space

This article is one of Manifold, Differential Geometry, Fibre Bundle.

3.1 Principal, Mean, and Gaussian Curvatures

Recall that a regular submanifold of a manifold $\tilde{M}$ is a subset of the manifold $\tilde{M}$ locally defined by the vanishing of coordinate functions. By a surface in $\mathbb{R}^3$, we mean a 2-dimensional regular submanifold of $\mathbb{R}^3$. Let $p$ be a point on a surface $M$ in $\mathbb{R}^3$. A normal vector to $M$ at $p$ is a vector $N_p\in T_p\mathbb{R}^3$ that is orthogonal to the tangent plane $T_pM$, A normal vector field on $M$ is a function $N$ that assigns to each $p\in M$ a normal vector $N_p$ at $p$. If $N$ is noraml vector field on $M$, the nat each point $p\in M$, we can write $$N_p=\sum_{i=1}^3 a^i(p)\left. \frac{\partial}{\partial x^i}\right|_p.$$ The noraml vector field $N$ on $M$ is said to be $C^\infty$ if the coefficient functions $a^1,a^2,a^3$ are $C^\infty$ functions on $M$.

Let $N$ be a $C^\infty$ unit normal vector field on neighborhood of $p$ in $M$. Denote by $N_p$ the unit normal vector at $p$. Under the canonical identification of $T_p\mathbb{R}^3$ with $\mathbb{R}^3$, every plane $P$ through $N_p$ slices the surface $M$ along a plane curve $P\cap M$ through $p$. By the transversality theorem from differential topology, the intersection $P\cap M$, being transversal, is smooth.(???) We call such a plane curve a normal section of the surface through $p$ (Figure 3.1).

Assuming that the noraml sections have $C^\infty$ parametrizations, which we will show later, we can compute the curvature of a normal section with respect to $N_p$. The collection of the curvatures at $p$ of all the noraml sections gives a fairly good picture of how the surface curves at $p$.

More precisely, each unit tangent vector $X_p$ to the surface $M$ at $p$ determines together with $N_p$ a plane that slices $M$ along a normal section. Moreover, the unit tangent vector $X_p$ determines an orientation of the normal section. Let $\gamma(s)$ be the arc length parametrization of this normal section with initial point $\gamma(0)=p$ and initial vector $\gamma'(0)=X_p$. Note that $\gamma(s)$ is completely determined by the unit gangent vector $X_p$. Define the normal curvature of the normal section $\gamma(s)$ at $p$ with respect to $N_p$ by $$\kappa(X_p)=\left\langle \gamma''(0),N_p\right\rangle.$$ Of course, this $N_p$ is not always the same as the $\mathbf{n}(0)$ in Section 2, which was obtained by rotating the unit tangent vector $90^\circ$ counterclockwise in $\mathbb{R}^2$; an arbitrary plane in $\mathbb{R}^3$ does not have a preferred orientation.

Since the set of all unit vectors in $T_pM$ is a circle, we have a function $\kappa:S^1\rightarrow \mathbb{R}.$ Clearly, $\kappa(-X_p)=\kappa(X_p)$ for $X_p\in S^1$, because replacing a unit tangent vector by its negative simply reverses the orientation of the normal section, which reverses the sign of the first derivative $\gamma'(s)$ but does not change the sign of the second derivative $\gamma''(s)$.

The maximum and minimum values $\kappa_1,\kappa_2$ of the function $\kappa$ are the principal curvatures of the surface at $p$. Their average $(\kappa_1+\kappa_2)/2$ is the mean curvature $H$, and their product $\kappa_1\kappa_2$ the Gaussian curvature $K$. A unit direction $X_p\in T_pM$ along which a principal curvature occurs is called principal direction. Note that if $X_p$ is a principal direction, then so is $-X_p$, since $\kappa(-X_p)=\kappa(X_p)$. If $\kappa_1$ and $\kappa_2$ are equal, then every unit vector in $T_pM$ is a pricipal direction.

3.2 Gauss's Theorema Egregium

$\kappa_1$, $\kappa_2$, and $K$ are invariant under isometries.

3.3 The Gauss-Bonnet Theorem

For an oriented surface $M$ in $\mathbb{R}^3$ the Gaussian curvature $K$ is a function on the surface. If the surface is compact, we can integrate $K$ to obtain a single number $\int_M KdS$. Here the integral is the ususal surface integral from vector calculus.

Gauss-Bonnet theorem. For compact oriented surface $M$ in $\mathbb{R}^3$ asserts that $$\int_M KdS=2\pi \chi(M),$$ where $\chi(M)$ denotes the Euler characteristic. Equation is a rather unexpected formula, for on the left-hand side the Gaussian curvature $K$ depends on a notion of distance, but the right-hand side is a topological invariant, independent of any Riemannian metric. Somehow, in the integration process, all the metric information gets canceled out, leaving us with a topologucal invariant. In due course we will study the theory of characteristic classes for vector bundles, a vast generalization of the Gauss-Bonnet theorem.