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ELEMENTARY DIFFERENTIAL GEOMETRY AND THE
GAUSS-BONNET THEOREM
DUSTIN BURDA
Abstract. In this paper we discuss examples of the classical Gauss-Bonnet
theorem under constant positive Gaussian curvature and zero Gaussian cur-
vature. We then develop the necessary geometric preliminaries with example
calculations. We then use our newly developed tools to prove the local Gauss-
Bonnet theorem. After a foray into triangulation and the Euler characteristic,
we will use the local Gauss-Bonnet theorem to prove the global Gauss-Bonnet
theorem for compact surfaces.
Contents
1. Introduction 1
2. Gauss-Bonnet for particular triangles 2
2.1. Zero Gaussian Curvature 2
2.2. Constant Positive Gaussian Curvature 3
3. Geometric Preliminaries 3
3.1. Curves 3
3.2. Surfaces 4
3.3. The Tangent Plane 5
3.4. The First Fundamental Form 5
3.5. The Gauss Map and the Second Fundamental Form 7
3.6. Curvature(s) 8
3.7. The Covariant Derivative and Geodesics 11
3.8. Three Lemmas 12
4. Local Gauss-Bonnet Theorem 14
5. Global Gauss-Bonnet Theorem 15
Acknowledgments 17
References 17
1. Introduction
The Gauss-Bonnet theorem serves as a fundamental connection between topol-
ogy and geometry. It relates an inherently topological quanitity of a surface, the
Euler characteristic, with an intrinsic geometric property, the total Gaussian cur-
vature. Gaussian curvature is, roughly speaking, how much a surface deviates from
a tangent plane at a point. The flatter the surface, the closer a tangent plane
approximates it, and the lower the Gaussian curvature. The Gauss-Bonnet theo-
rem remarkably states that under homeomorphism, the total Gaussian curvature
remains invariant. This seems quite odd upon first glance as twisting, squishing, or
1
2 DUSTIN BURDA
stretching an object seems like it would radically alter many of its geometric prop-
erties. Yet, since the Euler characteric remains invaraiant under homeomorphism,
the Gauss-Bonnet theorem predicts that total curvature should remain unchanged.
Our goal is to build the necessary machinery to understand the theorem and then
prove it. It formally states:
Theorem 1.1. Let R ⊂ S be a regular region of an oriented surface and let C1
· · · Cn be the closed, simple, piecewise regular curves which form the boundary ∂R
of R. Suppose that each C is positively oriented and let θ ··· θ be the set of all
i 1 p
external angles of the curve C1 ··· Cn . Then
n Z ZZ p
Xcikg(s)ds+ RKdσ+Xθl=2πχ(R),
i=1 l=1
where s denotes the arc length of C , and the integral over C means the sum of
i i
integrals in every regular arc of Ci.
χ refers to the Euler characteristic and K refer to the total Gaussian curvature.
The importance of Gauss-Bonnet is the connection it makes between these seem-
ingly unrelated values. We will motivate both of these notions later in the paper
and supply the requisite definitions.
2. Gauss-Bonnet for particular triangles
Thefirstversion of the Gauss-Bonnet theorem that we will discuss concerns itself
with geodesic triangles on a surface. It states that the difference between the angles
of the triangle and π equals to the total Gaussian curvature of the triangle or
3 ZZ
(2.1) Xφi−π= TKdA.
i=1
Geodesics, roughly speaking, are the shortest path along a surface. On a plane
they are lines, on a sphere they are great circles, etc. We can think of curvature
as how fast a tangent plane to a given point deviates from a surface. This formula
enables us to compute the total Gaussian curvature along the triangle in a relatively
simple manner. As opposed to complex computation with integrals, we calculate
the angles in a given geodesic triangle.
2.1. Zero Gaussian Curvature. In a plane, the value K, or the Gaussian curva-
ture, is equal to 0. This implies that the value on the right side of (2.1) evaluates
to 0. Additionally, geodesics along the plane are merely straight lines. This im-
plies that a geodesic triangle in a surface of flat curvature is a planar triangle.
Rearranging the terms yields
3
Xφ =π.
i
i=1
This reduces to the theorem that the sum of the angles in a planar triangle add up
to π radians.
ELEMENTARY DIFFERENTIAL GEOMETRY AND THE GAUSS-BONNET THEOREM 3
2.2. Constant Positive Gaussian Curvature. Our next example is the unit
sphere, which has constant positive curvature. The Gaussian curvature of a unit
sphere is 1, meaning the right side of (2.1) is
(2.2) ZZT 1dA,
but this is the area of the region we are integrating, which in our case is the geodesic
triangle. Hence it is sufficient to prove
3
(2.3) Xφ −π=△,
i
i=1
where △ refers to the area of the geodesic triangle.
Proof. Suppose there exists a geodesic triangle with angles α, β, γ. For each pair
of edges of the triangle, their extensions along great circles form a lune, the 3 di-
mensional analogue of a sector of a circle. The surface area of a lune is proportional
to the angle it makes divided by 2π. In this case, the surface area of a lune with
angle α is α S, where S is the surface area of the sphere. Now, for each angle in the
2π
geodesic triangle, there exists two corresponding lunes. Hence, there are 6 lunes
total. Their union is the surface area of the sphere. However, the geodesic triangle
and its antipodal image are overcounted an additional 2 times by each lune, thus
the geodesic triangle is overcounted 4 times. We then obtain the following equation:
2( α + β + γ )S =S+4∆.
2π 2π 2π
However, S = 4π, the surface area of a unit sphere. Therefore, substitution and
simplification yields:
α+β+γ−π=∆.
3. Geometric Preliminaries
3.1. Curves. Before we prove the local and global Gauss-Bonnet theorems, we
need to construct some objects and the operations we wish to perform on them.
Definition 3.1. A parametrized differentiable curve is a differentiable curve
α: I → R3 of an open interval I = (a,b) of the real line R into R3.
Remark 3.2. For each t ∈ I, we denote α at t by α(t) = (x(t), y(t), z(t)) for some
parametrization of the curve.
Definition 3.3. The tangent vector α’(t) is the vector (x′(t), y′(t), z′(t))
Example 3.4. The map α(t) = (cos2t, sin4t, t2), where t ∈ R is a parametrized
differentiable curve. Each component is differentiable and α is a map from R into
R3.
The tangent vector at a point t is α′(t) = (−2sin(2t),4cos(4t),2t)
Definition 3.5. A parametrized differentiable curve α: I → R3 is said to be regular
if α’(t) 6= 0 for all t ∈ I.
Avariety of analytic tools can be used to explore the geometry of curves:
4 DUSTIN BURDA
(1) The first derivative is the tangent vector.
(2) The magnitude of the second derivative is defined to be the curvature. It
measures the rate of the change of the tangent line.
(3) The plane determined by the unit vectors in the direction of the first and
second derivatives is the osculating plane at a point.
(4) The measure of how quickly neighboring osculating planes differ from a
given one is torsion.
Curvature and torsion roughly correspond to the bending and twisting of a
curve. Therefore we can think of any curve as bending, twisting, and stretching an
interval. These serve as examples of how we can use analysis to explore geometry.
Corresponding notions of bending and twisting as well as geometric properites such
as area and length will be constructed for regular surfaces using the same idea.
3.2. Surfaces. We will now examine the notion of a surface. Surfaces are defined
to be subsets rather than mappings.
Definition 3.6. A subset S ⊂ R3 is a regular surface if, for each p ∈ S, there exists
a neighborhood V in R3 and a map x: U → V ∩ S of an open set U ⊂ R2 onto V
∩ S such that:
(1) x is smooth.
(2) x is a homeomorphism .
(3) For each q ∈ U, the differential dxq: R2 → R3 is injective.
Definition 3.7. The mapping x is called a parametrization of p.
Definition3.8. TheneighborhoodV∩SofpinSiscalledacoordinate neighborhood.
In order to do differential geometry, we need all derivatives of x to be well-
defined. This is guaranteed by Condition 1. Condition 2 enables our surface to be
locally Euclidean, meaning it will inherit many of the same tools present in Eu-
clidean space when viewed in a sufficently small neighborhood. Condition 3 takes
onamorefamiliarformifwecomputeitsmatrixinthecanonicalbasisofR2 andR3.
Let dxq have coordinates (u,v) in R2 and (x,y,z) in R3. The vector e1 is tangent
to the curve u → (u, v ). It’s image under x is (x(u,v ),y(u,v ),z(u,v )). The
0 0 0 0
tangent vector to the curve at x(q) is
∂x = ∂x ∂y ∂z.
∂u ∂u ∂u ∂u
Thus, by the defintion of differential:
dxq(e1) = ∂x ∂y ∂z = ∂x,
∂u ∂u ∂u ∂u
hence
∂x ∂x
∂u ∂v
dx =∂y ∂y.
q ∂u ∂v
∂z ∂z
∂u ∂v
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