Provided we show that our definition does not depend on the particular choice of geodesic polar coordinate system, a consequence of the above definition is that the curvature is a smooth function onS, since on any strong normal neighbourhoodW of a pointP∈S, it is clearly smooth onW \ {P}.
Before going on to prove (ii), let us consider our definition in the case of the three classical geometries. ForR2andS2, we already know that our definition gives the correct answer for the curvature, as both are embedded surfaces. In the case of the disc model Dof the hyperbolic plane, for any given pointP ∈ D, we can find an isometry ofDwhich sendsPto the origin. Thus, the calculation we perform in (iii) below verifies that the curvature is constant with value−1.
Example
(i) ForR2, we haveρ=r, and the metric isdρ2+ρ2dθ2; thus√
G=ρandK=0.
(ii) ForS2, the metric isdρ2+sin2ρdθ2(with respect to local geodesic polar coordinates).
Therefore√
G=sinρ, and so(√
G)ρρ = −√
GandK=1.
(iii) On the disc model of the hyperbolic plane, the Riemannian metric is given (in geodesic polar coordinates) bydρ2+sinh2ρdθ2. Therefore√
G=sinhρ, and soK= −1.
8.3 Gauss–Bonnet for geodesic triangles
Let S denote an abstract smooth surface equipped with a Riemannian metric. For R ⊂ S a suitably well-behaved region and K a continuous function onR, we are able to form an integral
RK dA. This should be understood on appropriate charts with coordinates(u,v)— in the usual notation, it is just
K(EG−F2)1/2du dv, the integral being taken over the appropriate region inR2. For a region contained in the domains of two different charts onS, the transition function is (by definition) an isometry with respect to the Riemannian metrics, and the fact that the integral is well defined follows as in the proof of Proposition4.7. This enables us to define the integral, even ifRis not contained in the domain of a single chart. WhenK=1, we just recover the area ofR.
Regions over which we shall be particularly interested in integrating are (geodesic) polygons onS, defined precisely as in Definition3.4in the case ofS2andT. An even more particular case of interest for us will be the case ofgeodesic triangles, whose sides we shall assume (as in the spherical case) to have the property that they are
the unique curves of absolute minimum length joining the relevant vertices. In the spherical case, this definition would however include the case of the complement of a spherical triangle — we shall therefore usually impose a further condition on our geodesic triangles that they be contained in a strong normal neighbourhood of one of their vertices.
The angles of such a geodesic triangle are determined via the Riemannian metric on any suitable chart, by taking the inner-product of tangents to the sides (we note below that the angles are non-reflex); because of the compatibility conditions that are stipulated in the definition of a Riemannian metric on an abstract surface, this will not depend on the choice of chart.
Lemma 8.6 Suppose that W is a strong normal neighbourhood of A ∈ S, and B,C are distinct points of W\ {A}, such that the curveof absolute minimum length joining B to C lies in W\{A}. The vertices A,B,C then determine a (unique) geodesic triangle in W .
Proof Observe that for each pointP on, the (unique) geodesic ray fromA and passing thoughP intersects only atP. If it were to intersect at another point Q = P, then there would be two geodesics inW joining PtoQ(since clearly the radial segmentPQcannot form part of), contradicting the assumption thatW was astrongnormal neighbourhood. NowW is the image of some open ballBε=B(0,ε) under the parametrizationσ, obtained via normal coordinates as in Theorem7.13, where the geodesic rays fromAnow correspond inBεto the radial rays from the origin.
The geodesic curve inBεcorresponding toinW is then a curve, contained in the sector of angleα(< π) inB(0,ε)determined by the radial rays corresponding toAB andAC. This last statement follows fromW being astrongnormal neighbourhood: if the curve were contained in the complementary sector with reflex angle 2π−α, then it would intersect thediametercorresponding toABtwice (once on either side of the origin), and hence it and the diameter would be different geodesics inBεjoining the same two points.
The radial geodesic segmentsAB andAC are absolutely length minimizing by Corollary7.18, as by assumption is the geodesic segmentBC. The concatenation of the three geodesic segmentsAB,BCandCAinWcorresponds to a simple closed curve inBεof a particularly accessible type — in particular, we know that its complement in R2has precisely two connected components, with the bounded one being contained inBε(see Exercise1.13). Moreover, the closure of this bounded component is the union of initial segments of radial rays, with arguments in a range[θ0,θ0+α], where αis the angle of the geodesic triangle atA— the image underσ of this set is the geodesic triangle inW we seek. The above description ensures that all three angles in the triangle are non-reflex (a fact we quoted above).
We assume now that a geodesic triangle =ABCis contained in a strong normal neighbourhood W of one of its vertices, say A. By considering geodesic polar coordinates(r,θ)onW, the Riemannian metric then takes the formdr2+G(r,θ)dθ2
8.3 GAUSS–BONNET FOR GEODESIC TRIANGLES 161 on the punctured neighbourhood. We may define a curvature functionKonW \ {A} by the formula from the previous section, namelyK:= −(√
G)ρρ/√ G.
Proposition 8.7 Suppose that a geodesic triangleis contained in a strong normal neighbourhood of one of its vertices, and has internal angles α,β and γ. If the curvature function K is defined as above on the punctured normal neighbourhood,
then
K dA=(α+β+γ )−π.
Proof The most direct proof of this is by explicitly integrating, in an analogous way to our proof in the case of a hyperbolic triangle. We shall take geodesic polar coordinates(r,θ)on the strong normal neighbourhood, and denote byσ :V → U the corresponding parametrization of an open subsetU ⊂S, withVan open subset of R2of the form(0,δ)×(λ,λ+2π). The sides of the triangle containingAcorrespond to taking constant values of theθcoordinate, which we may assume to beθ=0 and θ =α, where we assumeλhas been chosen so that[0,α] ⊂(λ,λ+2π). There is a very minor objection to taking such a parametrization by geodesic polars, in that its image does not include the vertexAof our triangle, but this may be got around by a suitable limiting argument, as the integrations we perform will not be affected by omitting the vertexA. The corresponding Riemannian metric, onV is then just dr2+G(r,θ)dθ2.
We observed above that, for each point P on the third side of the geodesic triangle, namely the sideBC, the (unique) geodesic ray fromAand passing throughP intersectsonly atP, and that the triangle is just the union of such geodesic segments fromAto a pointPof.
A
B C
u b
g c(u) P
We deduce therefore thatmay be parametrized by the coordinateθ, with(θ) = σ (f(θ),θ)representing the unique point of intersection of the geodesic rayγθ with , where 0≤θ≤α(withB=(0)andC=(α)). Here, the functionf(θ)is just the length of the radial geodesic fromAto the pointP=(θ). We letψ(θ)denote the angle, as shown in the diagram, at which the curvemeets the radial geodesic corresponding to a given value ofθ; in particular,ψ(0)=π−βandψ(α)=γ.
We now work entirely on the chart V. The three sides of the triangle are then given by the rays θ = 0 and θ = α, and the curve η(θ) = (f(θ),θ). If we let s denote the arc-length of η, then ds/dθ = η(θ) = h(θ) say, where h(θ) =
f(θ)2+G(f(θ),θ)1/2
> 0. Here, as in the rest of the proof, we shall use the prime notation for differentiation with respect to the parameter θ. If we reparametrize ηin terms of its arc-length, then it satisfies the geodesic equations (7.3); the first of these is 2d2f/ds2=Gr(dθ/ds)2, or in terms ofθ-derivatives that
1 h
1 hf
= 1 2h2Gr. We know thate1ande2/√
Gform an orthonormal basis forR2with respect to the metric, and thatη(θ) =(f(θ), 1)has normh(θ). We may identify the angleψby means of the relation cosψ= e1,η/h=f/h. We also have the relation
h√
Gsinψ= e2,η = e22=G, that is sinψ = √
G/h. Differentiating the first of these relations with respect tos gives−ψsinψ/h=(f/h)/h=Gr/2h2, using the above geodesic condition, and therefore (using the second of the relations to substitute for sinψ) that
ψ= −1 2Gr/√
G= −(√ G)r. Using the formulaK = −(√
G)rr/√
Gfor the curvature function, the integral we want is
K dA= α
0
f(θ)
0
K√
G dr dθ= − α
0
f(θ)
0
(√
G)rrdr dθ.
We integrate with respect tor, and use the relationψ(θ) = −(√
G)r(f(θ),θ)and the fact (see Remark7.17) that(√
G)r →1 asr→0, obtaining
K dA= α
0
(ψ+1)dθ=γ−(π−β)+α,
as required.
We can now show that the curvature takes a well-defined value, independent of choices made — in doing so, we shall recover an equivalent, but far more geometric, definition for the curvature in Definition8.8.
Suppose thatW is a strong normal neighbourhood of some pointP ∈ S. With respect to the geodesic polar coordinates(r,θ)onW, the Riemannian metric then takes the formdr2+G(r,θ)dθ2onW \ {P}, and we define a smooth functionKon W \ {P}byK:= −(√
G)ρρ/√ G.
Let us take any point Q ∈ W \ {P}. We shall show in Lemma 8.13 that, for sufficiently smallε > 0, the geodesic ball U with centre Qand radius ε has the
8.3 GAUSS–BONNET FOR GEODESIC TRIANGLES 163 property that any three distinct points A,B,C of U determine a unique geodesic triangle inU; we assume also thatεis chosen withU ⊂W \ {P}. Moreover, such a triangle may be expressed in terms of unions and set-theoretic differences of the three geodesic trianglesPAB,PBCandPCAinW, as shown in the diagram below (the third case illustrates the possibility thatPBC may degenerate, and so only two triangles are involved). The main reason for having taken astrongnormal neighbourhoodW was to ensure that these triangles exist (Lemma8.6).
P P P
A B
C
A
B
A
B C C
We can apply Proposition8.7to each of the geodesic trianglesPAB,PBC andPCA;
by taking appropriate sums and differences of the resulting formulae, we deduce that
K dA=(α+β+γ )−π, whereα,βandγ denote the internal angles of. If nowU is a neighbourhood ofQof the above type, contained in two different punctured strong normal neighourhoodsW1\ {P1}andW2\ {P2}ofP1(respectively P2), andK1(respectively K2) denote the corresponding curvature functions onU, then the above argument shows that, for any triangle ⊂U,
K1dA =
K2dA.
If we now choose smaller and smaller triangles ⊂U whichcontainthe pointQ (for instance, withQa vertex), we may deduce thatK1(Q)=K2(Q). This was the second claim made in the previous section, that the curvature we had defined did not depend on the choice of geodesic polar coordinate system.
Moreover, we see that the somewhat opaque definition of curvature we made above is equivalent to a very attractive geometric definition of curvature, namely:
Definition 8.8 The curvatureK at any pointQ ∈ S may be recovered from the geodesic trianglesof small diameter containingQ, by the formula
K = lim
diam→0
angles of −π
area .
Having shown now that the curvature function is both well defined and well behaved, we have other equivalent definitions, which provide further insight. LettingPdenote a point on a smooth abstract surface S, equipped with a Riemannian metric, we have local geodesic polar coordinates (ρ,θ), with respect to which the metric is dρ2 +G(ρ,θ)dθ2. By Lemma7.16,√
G(ρ,θ)is a smooth function ofρ andθ
(see also Remark7.17); assuming for simplicity that it may be expanded as a power series inρ with coefficients depending onθ (the general case is a straightforward extension, which is left to the reader), the facts from Remark7.17that√
G→0 and (√
G)ρ → 1 asρ → 0, together with the characterization of the curvature atP as K =limρ→0(√
G)ρρ/√
G, ensure that√
Gis locally of the form G(ρ,θ)=ρ (1−Kρ2/6 + higher order terms inρ ).
Note that we have managed to find two further terms in the expansion of√ G, as compared to Remark7.17.
If we now take a small geodesic circle of radius ε with centre at P, with circumferenceC(ε)and areaA(ε), one checks easily that we may recover the curvature Kas the limits asε→0 of both
3(2πε−C(ε))/πε3 and 12(πε2−A(ε))/πε4.
The geometric characterizations of curvature which we have produced in the last few paragraphs are generalizations of calculations we made in the specific cases of the sphere and hyperbolic plane.
Remark With the curvatureKnow shown to be well defined, Proposition8.7is just the Gauss–Bonnet formula for geodesic triangles of the form being considered.
This will be a crucial ingredient in our proof of the global Gauss–Bonnet theorem, which we prove in the following section.
Finally in this section, we mention a couple more basic results on curvature. The first of these describes how the curvature behaves under rescaling the metric by a positive real numberc2; for instance, a sphere of radiusc>0 has curvature 1/c2.
Lemma 8.9 Suppose S is a surface equipped with a Riemannian metric g with curvature K . If we consider the scaled metric c2g on S, then the curvature becomes K/c2.
Proof This follows most easily from our definition of curvature in terms of small geodesic triangles containing a given point Q. Note that scaling the metric does not change angles, and so for a given small trianglecontainingQ, we have that
K dAis invariant under the scaling. Since the area ofscales byc2, the claim
follows directly from the above definition.
Suppose we have an abstract smooth surface equipped with a Riemannian metric, which is locally isometric to the spherical metric, the Euclidean metric or the hyperbolic metric. This then ensures that the Gaussian curvature onS is constant, namely 1, 0 or−1 respectively. We can now prove the converse to this, a classical result due to Minding, that a surface with a constant curvature metric is (after rescaling the metric) locally isometric to an open subset of one of the three basic classical geometries.