Math 4530 Supplement. April 23, 2004

Geodesics and the Gauß-Bonnet Theorem

Andrejs Treibergs

Abstract. In these notes we compute the geodesic curvature on a surface in isothermal coordinates and use it to prove the local Gauß-Bonnet Theorem.

These remarks are a continuation of my notes [T] whose notation we continue to employ.

1. Isothermal Coordinates of a Surface.

The computations are facilitated by using a special coordinate system in which the metric and the resulting formulas take a particularly simple form.

Theorem [Isothermal Coordinates]. [Korn-Lichtenstein 1914, Lavrentiev, Morrey] Suppose M is a surface of class C^{k,α} (k ∈ N, 0 < α < 1) or C^{∞} and P ∈ M. Then another coordinate patch X : Ω → M with P ∈ X(Ω) may be found (Ω is an open set of ℝ^{2}) such that X(u^{1}, u^{2}) ∈ C^{k,α}(Ω) or C^{∞}(Ω), resp., such that the conformality relations g_{11} = X_{1} · X_{1} = g_{22} = X_{2} · X_{2} > 0 and g_{12} = X_{1} · X_{2} = 0 are satisfied for all (u^{1}, u^{2}) ∈ Ω.

For a proof of this, see Jost [J]. Thus the metric g_{11} = g_{22} = φ(u^{1}, u^{2}) is given by a single function. The Gauss Curvature formula for orthogonal coordinates reduces to

(1)  K = −\frac{1}{2φ}Δ log φ.

where Δ = \frac{∂^{2}}{∂(u^{1})^{2}} + \frac{∂^{2}}{∂(u^{2})^{2}} is the usual Laplace operator.

We shall have occasion to use the Christoffel symbols associated to this metric.

(2)  Γ_{ij}^{m} = \frac{1}{2}g^{ℓm} \{\frac{∂}{∂u^{j}} g_{iℓ} + \frac{∂}{∂u^{i}} g_{ℓj} − \frac{∂}{∂u^{ℓ}} g_{ji}\} .

Let us assume that g_{12} = 0 and g_{11} = g_{22} = φ on all of our coordinate patch Ω. Thus also g^{ij} = δ^{ij}/φ. Then using (2), the Christoffel symbols take the form

(3)  \begin{matrix}Γ_{11}^{1} = & \frac{φ_{1}}{2φ} \\ Γ_{21}^{1} = Γ_{12}^{1} = & \frac{φ_{2}}{2φ} \\ Γ_{22}^{1} = & − \frac{φ_{1}}{2φ} \end{matrix}  \begin{matrix}Γ_{11}^{2} = & − \frac{φ_{2}}{2φ} \\ Γ_{21}^{2} = Γ_{12}^{2} = & \frac{φ_{1}}{2φ} \\ Γ_{22}^{2} = & \frac{φ_{2}}{2φ}. \end{matrix}

2. Geodesic Curvature of a curve on the surface a Surface.

Consider a curve γ(s) ∈ M in the patch parameterized by arclength. Then, the tangent vector T = γ′ is a unit vector tangent to M where “′ ” denotes \frac{d}{ds}. The surface normal vector U is another unit vector orthogonal to T. Following the text, we denote the third vector W = U × T so that {T, W, U} is a moving orthonormal (right handed) frame along γ adapted to the surface. The normal curvature and the geodesic curvature are the normal and tangent components of T′. We considered such a frame when we discussed White’s formua relating the linking number to the twisting number of a pair of closed curves [O, p. 24]. Starting with orthonormality T · T = W · W = U · U = 1 and T · W = T · U = W · U = 0, by differentiating we find generalized Frenet equations for the accelerations

\{ \begin{matrix}T′ = & + ω_{3}W − ω_{2}U \\ W′ = & − ω_{3}T  + ω_{1}U \\ U′ = & ω_{2}T − ω_{1}W \end{matrix}

where the ω_{i} are functions. If we regard γ ∈ M ⊂ ℝ^{3} as a space curve, then we may suppose that it has the usual orthonormal Frenet Frame {T, N, B} moving with the curve. Since N is a unit vector perpendicular to T, we may write

(4)  N = cos(θ) W + sin(θ) U

where the function θ = ∠(N, W) gives the angle between M and W. The Frenet equation defines the curvature κ and the torsion τ of the space curve by the formulæ

\{ \begin{matrix}T′ = & + κN \\ N′ = & − κT  + τB \\ B′ = & − τN \end{matrix}

The normal curvature is given in terms of the shape operator, thus differentiating U · T = 0, and using T′ = κN,

k_{n} = S(T) · T = −U′ · T = U · T′ = U · κN = κ sin(θ) = −ω_{2}.

Let us define the geodesic curvature k_{g} as the other component

k_{g} = W · T′ = W · κN = κ cos(θ) = ω_{3}.

Thus the normal and geodesic curvatures satisfy the relation

k_{n}^{2} + k_{g}^{2} = κ^{2}.

It remains to determine ω_{1}. Using the right-handedness of the frame, T × W = U, W × U = T and U × T = W,

(5)  B = T × N = cos(θ) T × W + sin(θ) T × U = − sin(θ) W + cos(θ) U.

Inverting the relations (4) and (5) we find

W = cos(θ) N − sin(θ) B
U = sin(θ) N + cos(θ) B.

Differentiating using the Frenet equation yields

W′ = cos(θ) N′ − sin(θ) B′ + (− sin(θ) N − cos(θ) B) θ′
= cos(θ) (−κ T + τ B) + τ sin(θ) N − θ′ U
= − k_{g} T + (τ − θ′) U

whence ω_{1} = U · W′ = τ − θ′.

The generalized Frenet Equations become

\{ \begin{matrix}T′ = & + k_{g} W  + k_{n} U \\ W′ = & − k_{g} T  + (τ − θ′) U \\ U′ = & − k_{n} T − (τ − θ′) W \end{matrix}

The curve γ is called a geodesic, if it is as straight as possible on the surface, which means that there is no tangential acceleration, k_{g} = 0. Let us show that the only geodesics on the sphere 𝕊_{r}^{2} are great circles. Consider a sphere of radius r centered at the origin, and tahe the outward unit normal

\frac{1}{r}X = U.

A curve α(s) on 𝕊_{r}^{2} parameterized by arclength satisfies α·α = r^{2}. Differentiating, α·T = 0 and T ·T +κα·N = 0 which implies that the curvature κ > 0 is nonzero. Since we are assuming k_{g} = 0 for α, its acceleration is purely in the normal direction, or

κN = T′ = k_{n}U = \frac{k_{n}}{r} X

Thus κ ≠ 0 implies N = fX for some function f. Thus since the lengths are known, f = ±\frac{1}{r} which is constant along α. Differentiating again using both Frenet systems,

−κT + τB = N′ = fT

from which it follows that τ = 0 and κ = \frac{1}{r}. However a space curve with τ = 0 is planar and with κ = \frac{1}{r} is a circle of radius r. Thus α is a great circle of 𝕊_{r}^{2}.

The tangential part of the motion of the curve is intrinsic to the surface, thus it may be computed strictly from the curve and the first fundamental form and its derivatives. Recall the acceleration formulæ in arbitrary coordinates.

\begin{matrix}U_{i} = & − h_{ij} g^{jk} X_{k}, \\ X_{ij} = & Γ_{ij}^{k} X_{k} + h_{ij}U, \end{matrix}

A curve on the surface may be given by

γ(s) = X(u^{1}(s), u^{2}(s))

Its velocity and acceleration is thus by the chain rule and acceleration formulæ

(6)  T =γ′ = X_{i}u̇^{i},
T′ =X_{i}ü^{i} + X_{ij}u̇^{i} u̇^{j} = (ü^{k} + Γ_{ij}^{k}u̇^{i} u̇^{j}) X_{k} + (h_{ij} u̇^{i} u̇^{j}) U

where now “˙ ” denotes \frac{d}{ds}. In our notation, the geodesic curvature and the normal curvature are given by

k_{n} =T′ · U = h_{ij} u̇^{i} u̇^{j}
(7)  k_{g} =T′ · W = (ü^{k} + Γ_{ij}^{k}u̇^{i} u̇^{j}) X_{k} · W

It follows that if the curve γ is a geodesic, k_{g} = 0, and since T′ · T = 0, then the equations of a geodesic curve are given by

(8)  ü^{k}(s) + Γ_{ij}^{k}[u^{1}(s), u^{2}(s)] u̇^{i}(s) u̇^{j}(s) = 0  for k = 1, 2.

This is a second order nonlinear 2 × 2 system of ordinary differential equations on the surface, involving only the first fundamental form.

Suppose that γ(σ) = X(u^{1}(σ), u^{2}(σ)) is a curve which is not necessarily given with arclength parameter. Then |X_{i}u̇^{i}|^{2} = g_{ij} u̇^{i} u̇^{j} so

T =\frac{γ_{σ}}{|γ_{σ}|} = \frac{X_{i}u̇^{i}}{|X_{i}u̇^{i}|} = \frac{X_{i}u̇^{i}}{|X_{i}u̇^{i}|},
\frac{dσ}{ds} =\frac{1}{\sqrt{g_{ij} u̇^{i} u̇^{j}}}

T′ =Ṫ \frac{dσ}{ds} = \frac{X_{i}ü^{i} + X_{ij}u̇^{i} u̇^{j}}{\sqrt{g_{ij} u̇^{i} u̇^{j}}} − \frac{\frac{d}{dσ} [g_{ij} u̇^{i} u̇^{j}] T}{2g_{ij} u̇^{i} u̇^{j}}

=\frac{(ü^{k} + Γ_{ij}^{k}u̇^{i} u̇^{j}) X_{k} + (h_{ij} u̇^{i} u̇^{j}) U}{\sqrt{g_{ij} u̇^{i} u̇^{j}}} − (\frac{d}{dσ} log \sqrt{g_{ij} u̇^{i} u̇^{j}}) T

where in this equation, “˙ ” denotes \frac{d}{dσ}. Since (7), the equations for a geodesic continue to be (8), even if s is not arclength and g_{ij} is any metric.

For example, in the Euclidean plane, the metric g_{ij} = δ_{ij} are constant functions so that Γ_{ij}^{k} = 0 and the geodesic equations reduce to ü^{i} = 0. In other words u^{i} = a^{i}s + b^{i} for constants a^{i}, b^{i}, which are linear functions. Thus the geodesics are straight lines.

In principle, one can work out W in terms of X_{i}u̇^{i} and g_{ij} for to get an intrinsic expression for k_{g} for any metric. But it is easier to work in a nice coordinate system, thus now we assume that X(u^{1}, u^{2}) is an isothermal coordinate patch. In this case the X_{i} are orthogonal and since |X_{i}| = \sqrt{φ}, we have a new orthonormal basis given by

e_{i} = \frac{1}{\sqrt{φ}}X_{i}.

Thus we can write the two orthonormal tangent vectors of (6),

T = \sqrt{φ} u̇^{1}e_{1} + \sqrt{φ} u̇^{2}e_{2}, W = −\sqrt{φ} u̇^{2}e_{1} + \sqrt{φ} u̇^{1}e_{2} = −u̇^{2} X_{1} + u̇^{1} X_{2}

Substituting into (7), using X_{i} · X_{j} = φ δ_{ij},

k_{g} = (ü^{k} + Γ_{ij}^{k}u̇^{i} u̇^{j}) X_{k} · (−u̇^{2} X_{1} + u̇^{1} X_{2}) = −φ (ü^{1} + Γ_{ij}^{1}u̇^{i} u̇^{j}) u̇^{2} + φ (ü^{2} + Γ_{ij}^{2}u̇^{i} u̇^{j}) u̇^{1}

Finally, we substitute the expressions (3) for the Christoffel symbols,

u̇^{1}Γ_{11}^{2} − u̇^{2}Γ_{11}^{1} = − \frac{u̇^{1} φ_{2}}{2φ} − \frac{u̇^{2} φ_{1}}{2φ}
u̇^{1}Γ_{12}^{2} − u̇^{2}Γ_{12}^{1} = \frac{u̇^{1} φ_{1}}{2φ} − \frac{u̇^{2} φ_{2}}{2φ}
u̇^{1}Γ_{22}^{2} − u̇^{2}Γ_{22}^{1} = \frac{u̇^{1} φ_{2}}{2φ} + \frac{u̇^{2} φ_{1}}{2φ}

and find using the unit speed |γ′| = 1 so (u̇^{1})^{2} + (u̇^{2})^{2} = \frac{1}{φ}

k_{g} =φ (ü^{2} u̇^{1} − ü^{1} u̇^{2}) + φ (u̇^{1}Γ_{ij}^{2} − u̇^{2}Γ_{ij}^{1}) u̇^{i} u̇^{j}
=φ(ü^{2} u̇^{1} − ü^{1} u̇^{2}) − \frac{1}{2} (u̇^{1}φ_{2} + u̇^{2}φ_{1}) (u̇^{1})^{2} + (u̇^{1}φ_{1} − u̇^{2}φ_{2}) u̇^{1} u̇^{2} + \frac{1}{2} (u̇^{1}φ_{2} + u̇^{2}φ_{1}) (u̇^{2})^{2}
=φ (ü^{2} u̇^{1} − ü^{1} u̇^{2}) − \frac{1}{2} (φ_{2} (u̇^{1})^{3} − φ_{1} (u̇^{1})^{2} u̇^{2} + φ_{2} u̇^{1} (u̇^{2})^{2} − φ_{1} (u̇^{2})^{3})
=φ(ü^{2} u̇^{1} − ü^{1} u̇^{2}) + \frac{1}{2} [φ_{1} u̇^{2} − φ_{2} u̇^{1}] ((u̇^{1})^{2} + (u̇^{2})^{2})
=\frac{ü^{2} u̇^{1} − ü^{1} u̇^{2}}{(u̇^{1})^{2} + (u̇^{2})^{2}} + \frac{φ_{1} u̇^{2} − φ_{2} u̇^{1}}{2φ}

Finally, if we denote the angle that the curve makes relative to the (u^{1}, u^{2}) coordinates by η, that is (cos η, sin η) = \sqrt{φ} (u̇^{1}, u̇^{2}), we find

(8)  k_{g} = η′ + \frac{1}{2} [(log φ)_{1} u̇^{2} − (log φ)_{2} u̇^{1}]

where we have differentiated η = tan^{−1} (\begin{matrix} & u̇^{2} \\ & u̇^{1} \end{matrix} ). Oprea [O, p. 308] deduces the same formula for an orthogonal

coordinate system.

3. Gauß Bonnet Formula closed curves in a coordinate patch.

Suppose γ is a closed simple curve in an isothermal coordinate patch. Assume that all of the interior region bounded by the curve is included in the patch such that the curve can be continuously contracted to a point through curves in the patch. We shall call such curves simply connected. A picewise smooth curve may have finitely many kinks. At each kink γ(s_{i}), where s_{1} < s_{2} < ⋯ < s_{n} are the nonsmooth points along the curve, let α_{i} denote the exterior angle. That is α_{i} = ∠(γ′(s_{i} + 0), γ′(s_{i} − 0)) is the angle from the ending tangent direction of the smooth curve before s_{i} on [s_{i−1}, s_{i}] to the starting tangent direction of the smooth curve after s_{i} on [s_{i}, s_{i+1}]. One of the advantages of the coordinates being isothermic is that the angle between two vectors in the (u^{1}, u^{2}) plane is the same as the angle between corresponding vectors on the surface M. Thus α_{i} = η(s_{i} + 0) − η(s_{i} − 0) is also the jump in the angle along the curves in the (u^{1}, u^{2}) plane at s_{i}.

Theorem. Let γ be a piecewise smooth, regular, simple, simply connected curve in an isothermal coordinate patch. Let D be the region inside the patch whose boundary is γ = dD. Then

∫∫_{D} K dA + ∮_{γ} k_{g} ds + ∑_{i=1}^{n} α_{i} = 2π.

Proof. We apply the divergence theorem. Use the fact that the area form on the surface is given by dA = \sqrt{g_{11}g_{22} − g_{12}^{2}} du^{1} du^{2} = φ du^{1} du^{2}. Also, the outside unit normal vector for a curve (u^{1}(s), u^{2}(s)) going around D in the counterclockwise direction is given by ν = \sqrt{φ}(u̇^{2}, −u̇^{1}) and the arclength in the (u^{1}, u^{2}) plane is ds/\sqrt{φ}. Thus we find from (1) and (8) that

∫∫_{D} K dA = − ∫∫_{D} (\frac{Δ log φ}{2φ}) φ du^{1} du^{2} = −\frac{1}{2} ∮_{γ} ∇ log φ · ν \frac{ds}{\sqrt{φ}}
= − \frac{1}{2} ∮_{γ} u̇^{2}(log φ)_{1} − u̇^{1}(log φ)_{2} ds = ∮ η′ − k_{g} ds

= − ∮ k_{g} ds + 2π − ∑_{i=1}^{n} α_{i}.

The last inequality follows from fact that the total angle η(L) − η(0) turned by a simple, simply connected curve is 2π. Here L is the length of γ. The contribution to the total turning at each corner is given by α_{i}. One can imagine approximating the curve near a corner by a very tight, yet smooth curve that is rounded at that corner, and which turns an angle α_{i} near the corner. The fact that the total turn of a piecewise regular simple, simply connected closed curve in the (u^{1}, u^{2}) plane is 2π is known as the Umlaufsatz, [dC, p. 396.] The integral of geodesic curvature plus the total of the exterior angles is the total turning 2π.

By covering a compact surface with piecewise smooth regions from isothermal charts, the global Gauß-Bonnet formula may be proved as in the text.

References

[dC] M. do Carmo, Differential Geometry of Curves and Suraces, Prentice Hall, Upper Saddle River, New Jersey, 1976.

[J] J. Jost, Harmonic maps between Surfaces, Springer-Verlag, Lecture Notes in mathematics, 1062, Berlin, 1984.

[O] J. Oprea, Differential Geometry and its Applications, 2nd. ed., Pearson Education Inc., Prentice Hall, Upper Saddle River, New Jersey, 2004.

[T] A. Treibergs, Gauß Curvature in Terms of the First Fundamental Form, University of Utah, Math 4530 Lecture Notes (March 26, 2004).

University of Utah, Department of Mathematics 155 South 1400 East, Rm 233 Salt Lake City, UTAH 84112-0090

E-mail address:treiberg@math.utah.edu