Let us consider first the familiar case of polar coordinates onR2 = C. Any non- zero point ofR2has unique polar coordinates(r,θ), withr >0 and 0≤ θ < 2π.
Moreover, given any angleθ0, there is a smooth parametrization ofU =C\R≥0eiθ0 σ :(0,∞)×(θ0,θ0+2π)→U
given byσ(r,θ)=(rcosθ,rsinθ), with inverse (chart) given locally by (x,y)→((x2+y2)1/2, tan−1(y/x)).
The second coordinate here needs to be interpreted appropriately, writing the function as cot−1(x/y)when x = 0, and choosing the correct value locally for tan−1(y/x), respectively cot−1(x/y). Note that the radial rays given by settingθconstant are just the geodesics starting at the origin, and that the radial coordinaterat a point is the distance along the relevant geodesic.
For an arbitrary surface, we can construct, at least locally, a similar coordinate system. In fact, it is sufficient to do this for an open subsetV ⊂R2equipped with a Riemannian metric. In the case of an embedded surface, we can find a chartψ:U → V withP ∈ U, andV an open subset ofR2equipped with the Riemannian metric given by the first fundamental form. In the case of an abstract surface, as introduced in the next chapter, such a chart to an open subsetV ofR2equipped with a Riemannian metric exists from the definition. The geodesics throughPonUwill then correspond underψto geodesics throughψ(P)inV, and the construction given below will yield immediately the required coordinate system on the corresponding open subset ofU.
We suppose therefore thatV ⊂R2is an open subset, equipped with a Riemannian metric, and assume thatP ∈ V. It follows from Proposition7.10that for any angle θ, there is a unique (germ of a) geodesic
γθ :(−ε,ε)→V
throughPwith ˙γθ(0) =1 and whose tangent atPhas polar angleθ. Recall from Remark7.7that geodesics have constant speed, and so ˙γθ(t) =1 for all−ε <t< ε.
7.5 GEODESIC POLARS AND GAUSS’ S LEMMA 145 We have already remarked that the geodesicγθ will depend smoothly on the initial parameterθ— rather more is however true.
Theorem 7.13
(i) For fixed P ∈ V , we may chooseε >0(independent ofθ) for which the geodesics γθ :(−ε,ε)→V are defined on(−ε,ε)for allθ. Moreover, if we vary P∈ V , we may takeεto be a continuous (in fact smooth) function of P.
(ii) Let Bεdenote theε-ball centred at the origin inR2, and define a mapσ :Bε →V byσ (rcosθ,rsinθ) := γθ(r)(withσ (0) = P). The mapσ is smooth, and, forε sufficiently small, it is a diffeomorphism from Bεonto an open neighbourhood of W of P in V .
Proof These results follow from an analysis of the differential equations involved, which the reader is invited to take on trust. Once we know that σ is smooth, we may proceed as follows: Givenv(θ)=(cosθ, sinθ)∈R2, we can consider the line segmentη(t)=tv(θ), for−ε <t< ε. By definition, we haveγθ(t)=σ (η(t)), and so by the Chain Rule
(dσ)0(v(θ))= ˙γθ(0)=0.
Since this holds for all θ, we have(dσ)0 is an isomorphism; the final claim then follows from the Inverse Function theorem. A full reference for these results is [12],
Chapter 9.
Definition 7.14 An open neighbourhood ofPof the formW =σ (Bε), as defined in Theorem 7.13(ii), is called a normal neighbourhood. We observe that, for any Q∈W \ {P}, there exists a unique geodesic inW fromPtoQ.
Choosing ε > 0 small enough therefore, we can define a smooth map g :(−ε,ε)×R→V by
g(r,θ):=γθ(r)=σ(rv(θ)),
wherev(θ) =(cosθ, sinθ). For any angleθ0, this restricts to a diffeomorphism of (0,ε)×(θ0,θ0+2π) onto an open subset of V. Note that the image of(0,ε)× (θ0,θ0+2π)undergis not a neighbourhood ofP. The image of(0,ε)×[θ0,θ0+2π) is the punctured neighbourhood W \ {P} of P, where W = σ (Bε)is the normal neighbourhood ofP. The coordinates(r,θ)constructed here are calledgeodesic polar coordinatesaroundP. Often one usesρinstead ofr, to indicate that this coordinate comes from the length of the geodesic ray. A point ofW\ {P}will have well-defined geodesic polar coordinates(r,θ)withr>0 and 0≤θ <2π.
With respect to geodesic polar coordinates, the geodesic rays inV starting from P correspond to takingθto be constant, and the ‘geodesic circles’ centred atP(of radius< ε) correspond to takingrto be constant.
Theorem 7.15 (Gauss’s lemma) The curves given by taking r to be a positive constant< εintersect all the geodesic rays through P at right angles.
Proof For fixedr< ε, consider the smooth curveαinW given by α(τ)=(α1(τ),α2(τ)):=σ(rcosτ,rsinτ).
For a given value ofτ, we have a geodesic rayστ(t):=σ (trcosτ,trsinτ), i.e. in our previous notationγτ(tr), where 0≤t≤1. Setting
h(t,τ):=σ(trcosτ,trsinτ)
for 0≤t≤1 andτ ∈R, we obtain a variation of the geodesicγ =σ0by geodesics στ which have fixed initial point P but variable end-point, namely α(τ). We let Q =α(0)=γ (1), the point where the geodesic rayγ meets the curveα. Eachστ has lengthrand constant speedr, so by Lemma7.8the energy ofστis also constant asτ varies.
a(t)
a(0) Q
s0 g st
P
a
Using the formula (7.2) for the derivation of the energy atτ =0, witha=0,b=1 andI =E(u,v)˙u2+2F(u,v)˙uv˙+G(u,v)˙v2, we deduce that
0=
E(Q)dγ1
dt (1) +F(Q)dγ2
dt (1) dα1
dτ (0) +
F(Q)dγ1
dt (1) +G(Q)dγ2
dt (1) dα2
dτ (0),
the two integrals in Equation (7.2) not appearing sinceγis a geodesic. This equation is just the statement
dγ dt (1),dα
dτ(0)
Q
=0,
which we needed to prove.
Given now the diffeomorphismσ : Bε → W as above, we defined a smooth map g:(−ε,ε)×R→W byg(t,θ)=σ (tv(θ)), wherev(θ)=(cosθ, sinθ)∈R2. This restricts to a smooth maph:(0,ε)×R→W, given by the same formula, and we have an induced Riemannian metric on(0,ε)×R, essentially just the Riemannian metric fromW expressed in terms of the geodesic polar coordinates(r,θ). We denote this metric byE dr2+2F dr dθ+G dθ2. Since the length of a geodesic ray is given by the
7.5 GEODESIC POLARS AND GAUSS’ S LEMMA 147 coordinater, it follows thatE=1. Gauss’s lemma says thatF =0. The Riemannian metric is therefore expressed, with respect to these geodesic polar coordinates, as
dr2 + G(r,θ)dθ2
The functionG(r,θ)is a smooth (positive) function on(0,ε)×R, periodic inθwith period 2π. Sinceh = (h1,h2)is locally an isometry (by construction), we deduce thatG(r,θ)= dh(e2)2= ∂h/∂θ2, wheree1,e2is the standard basis forR2, and the norm is determined by the Riemannian metric ath(r,θ). This however shows that we may extendGto a smooth (positive) function on(−ε,ε)×R, by setting
G(t,θ)= ∂g
∂θ(t,θ) 2
g(t,θ)
.
Lemma 7.16 With G(t,θ) the function on(−ε,ε)×R defined above, we have G(t,θ)=t2q(t,θ)for some positive smooth function q on(−ε,ε)×Rwith q(0,θ)=1 for allθ.
Proof Recalling thatg(t,θ)=σ(tv(θ))=γθ(t), we have
∂g
∂θ(t,θ)=(dσ )tv(tv(θ)), wherev(θ)=(−sinθ, cosθ)=v(θ+π/2). Therefore
G(t,θ)= ∂g/∂θ2=t2q(t,θ),
whereq(t,θ)= (dσ)tv(v(θ))2is a smooth function, strictly positive fort =0. To find the value ofqatt=0, we setφ=θ+π/2 and observe that
q(0,θ)= (dσ )0(v(φ))2P= ˙γφ(0)2=1.
Remark 7.17 Considering G(r,θ)for r > 0, this result determines the initial asymptotics ofGasr → 0. It follows immediately for instance thatG(r,θ) → 0 andGr/r → 2 asr → 0. Moreover, sincet q(t,θ)1/2is also a smooth function on (−ε,ε)×R, it follows that(√
G)r →1 asr→0.
We now observe that the above formula for the metric, in terms of geodesic polar coordinates, enables us to tie up a loose end from earlier in this chapter. The proof given shows that the result is equally true for geodesics on embedded (or, more generally, abstract) surfaces.
Corollary 7.18 With the notation as above, the radial geodesic curve from P to σ (r0,θ0) = Q of length r0 < εrepresents an absolute minimum for the length of curves joining P to Q. More generally, geodesics always locally minimize length and energy.
Proof The above formula for the metric in terms of geodesic polar coordinates enables us to estimate the length of curves from below. Suppose thatγ (t)is a smooth curve joiningPtoQ, which is contained inW =σ(Bε); then
lengthγ =
(˙r2+Gθ˙2)1/2dt ≥
|˙r|dt ≥r0,
with equality if and only ifθ˙=0 andr(t)is monotonic (cf. proof of Proposition5.8).
On the other hand, any smooth curve γ (t)joining PtoQ, which does not remain inW, must have length at leastε(cf. proof of Lemma4.4). Hence the geodesic ray segment fromPtoQgiven byθ=θ0represents an absolute minimum for the lengths of curves joiningPtoQ.
In general, we reduce locally to the case of the geodesic lying in a normal neighbourhood of its initial point, and hence representing an absolute minimum for
both the length and energy (using Lemma7.8).
Thus in Gauss’s lemma, the curves given by takingrto be a positive constant< ε consist of the points whose geodesic distance fromPisr, and so are indeedgeodesic circlescentred atP. We note in passing that geodesic circles (however parametrized) are usually not geodesics (Exercise7.7).
To illustrate the above ideas, we return to our three classical geometries, and calculate their metrics in terms of geodesic polar coordinates, which we shall now denote as(ρ,θ).
Example
(i) For the Euclidean plane,R2, the geodesic polar coordinates at the origin coincide with the standard polar coordinates(r,θ), and with respect to these coordinates the metric isdρ2+ρ2dθ2: thusG(ρ,θ)=ρ2.
(ii) For the sphere S2, with respect to geodesic polar coordinates at a point, which we may for instance take to be the north pole, the metric takes the form dρ2 + sin2ρ dθ2. For a formal proof of this, we can use the geodesic chart σ(ρ,θ) = (sinρ cosθ, sinρ sinθ, cosρ). ThereforeG(ρ,θ)=sin2ρ.
sin r r
7.5 GEODESIC POLARS AND GAUSS’ S LEMMA 149 (iii) On the disc model of the hyperbolic plane, the metric is
2 1−r2
2
(dr2+r2dθ2),
with respect to the standard polar coordinates. Consider now geodesic polar coordinates (ρ,θ) at the origin, where ρ = 2 tanh−1r. Therefore, dρ2 = 2
1−r2
2
dr2. Butr=tanh12ρ, and so( 4r2
1−r2)2 =sinhρ; thus the metric (in geodesic polar coordinates) isdρ2+sinh2ρdθ2. ThereforeG(ρ,θ)=sinh2ρ.
This yields yet another model of the (punctured) hyperbolic plane, namely R2 with Riemannian metric dr2+sinh2r dθ2, defined in terms of the standard polar coordinates; by changing coordinates, we see that this also defines a metric at the origin.
Finally, we prove an easy but important local result concerning the shape of geodesics in a normal neighbourhood of a point. Consider a normal neighbourhoodW =σ (Bδ) of a pointP; to fix ideas, we may as before reduce to the case whenP∈V, an open subset ofR2equipped with a Riemannian metric, although the argument is equally applicable to the case of an embedded or abstract surface. With respect to geodesic polar coordinates onW\ {P}, the metric may be expressed asdr2+G(r,θ)dθ2. We recall thatGr/r→2 asr→0.
Lemma 7.19 Suppose that Q1,Q2 ∈ W , and thatγ (t)is a geodesic segment in W joining the two points. If Gr >0at all points of this geodesic segment, then the maximum distance from P to a point on the geodesic is attained at either Q1or Q2. If Gr <0at all points of this geodesic, then the minimum distance from P to a point on the geodesic is attained at either Q1or Q2.
Proof SinceGr/r→2 asr→0, the geodesic segment will not containP.
We may assume therefore that the geodesic segment is contained inW \ {P}; we shall be interested in the local form ofγ, and so we can writeγin terms of geodesic polar coordinatesγ (t)=(r(t),θ(t)); thus|| ˙γ||2= ˙r2+Gθ˙2. The first of the geodesic equations is now of the form
¨ r= 1
2Grθ˙2.
So ift=t0represents a stationary point forr, we haver˙=0 andθ˙2>0. In the case whenGr >0, we cannot have a local maximum ofr(t)att =t0; whenGr <0, we cannot have a local minimum. The lemma therefore follows.
If one flies from New York to Moscow, along the (shorter) great circle route, one’s distance from the north pole is maximum at take-off, and minimum at some point during the journey. If, on the other hand, one flies from Rio de Janeiro to Sydney, Australia along the (shorter) great circle route, one’s distance from the north pole is minimum at take-off, and maximum at some point during the journey. This follows from the above calculation, together with a rudimentary knowledge of geography,
since if we take geodesic polar coordinates(ρ,θ)at the north pole, thenG(r,θ)= sin2ρ; thus Gρ = 2 cosρsinρ = sin 2ρ is positive in the northern hemisphere (ρ < π/2), and negative in the southern hemisphere. In this case however, a simple geometric argument (which we saw in the proof of Proposition2.16) also suffices to prove these facts.
Exercises
7.1 Show that any line on an embedded surfaceS ⊂R3must be a geodesic. Hence, find infinitely many geodesics on the hyperboloid of one sheet, with equationx2+y2= z2+1, in addition to those obtained via Proposition7.12.
7.2 Suppose we have a Riemannian metric on an open discDof radiusδ >0 centred on the origin inR2, possibly withDbeing all ofR2, given in standard polar coordinates by
(dr2+r2dθ2)/h(r)2,
whereh(r) >0 for all 0≤r< δ. Write down the geodesic equations for this metric.
Show that any radial curve, parametrized so as to have unit speed, is a geodesic.
7.3 With the Riemannian metric as in the previous question, show directly, without using the geodesic equations, that the length minimizing curves through the origin are just the line segments which contain the origin.
7.4 Fora>0, letS ⊂R3be the circular half-cone defined byz2=a(x2+y2),z>0, considered as an embedded surface. Show that the metric onS is locally Euclidean.
Whena=3, give an explicit description of the geodesics and show that no geodesic intersects itself. Fora >3, prove that there are geodesics (of infinite length) which intersect themselves.
7.5 LetS ⊂ R3 be an embedded surface andH ⊂ R3 a plane which is normal toS (i.e. contains the unit normal vector) at each point of the intersectionC =S ∩H.
Supposeγ is a constant speed curve onS whose image is contained inC; deduce from Proposition7.6thatγ is a geodesic onS.
7.6 Using Proposition7.6, provide an alternative proof of Proposition7.12.
7.7 LetV denote an open subset ofR2equipped with a Riemannian metric, and suppose that we have geodesic polar coordinates(r,θ)at a pointP∈V, forr< ε. For a fixed r0< ε, suppose that the functionG(r,θ)defined above hasGr(r0,θ)=0 for only finitely many 0≤θ <2π. Show that the geodesic circle centred onPwith radiusr0
is not a geodesic. Give an example of a geodesic circle which is a geodesic.
7.8 Consider the Riemannian metric on the unit disc inR2defined by 1
1−r2(dr2+r2dθ2),
with respect to the standard polar coordinates. Express the metric in terms of the corresponding geodesic polar coordinates (centred on the origin).
7.9 Letη(t) = (f(t), 0,g(t)) : [a,b] → R3be a smooth embedded curve in thexz- plane, withf(t) >0 for allt, and letS denote the surface of revolution defined by η, with boundary consisting of the two circlesC1andC2, with equationsz=g(a),
EXERCISES 151 x2+y2 = f(a)2, respectivelyz = g(b),x2+y2 = f(b)2. Suppose now thatη represents a stationary point for the area of S; ifg(a) = g(b), prove that S is a catenoid, as described in Section7.1. What happens ifg(a)=g(b)?
7.10 Suppose V is an open subset of R2, equipped with a Riemannian metric, whose associated metric space iscomplete. Show that any geodesicγ :(−ε,ε)→V may be extended to acompletegeodesic, that is a geodesicγ : R → V. Show that the same fact holds for complete embedded surfaces. [The converse is also true in both cases, and follows from theHopf–Rinow theorem.]
8 Abstract surfaces and Gauss–Bonnet