Calculating these dot products, we have K√
G=λ1à2−λ2à1
= −(√ G)uu
(the second equality coming from (∗∗)), and therefore we obtain the formula K= −(√
G)uu/√
G.
ForPa point on an embedded surfaceS ⊂R3, the local geodesic polar coordinates (ρ,θ)atP, and hence also the functionG(ρ,θ), depend only on the first fundamental form (i.e. the metric). The curvature on the corresponding coordinate patch onSis then given byK = −(√
G)ρρ/√
G. The pointPcorresponds toρ=0 and is technically not in the coordinate patch; in fact, we saw in Lemma 7.16that limρ→0G = 0.
However, all our functions, including the curvature, are smooth, and so the above equation also determines (in the limit asρ →0) the curvature atP. Thus we deduce the following corollary, a result of which even Gauss was rather proud, although this is not the proof he gave. Gauss himself used the adjective egregium (which translates intoremarkableoroutstanding) to describe this result.
Corollary 8.2 (Gauss’s Theorema Egregium) The curvature of an embedded surface depends only on the first fundamental form. In particular, if two embedded surfaces locally have isometric charts, then the curvatures are locally the same.
8.2 Abstract smooth surfaces and isometries
In Chapter 4, we studied arbitrary Riemannian metrics on open subsets ofR2. In Chapter 6, we studied embedded surfacesS inR3. These were covered by charts, identifying open subsets ofS homeomorphically with open subsets ofR2, and the identifications were consistent with the natural smooth structures on these open subsets of R2 by Proposition 6.2. However, these open subsets of R2 also carry Riemannian metrics corresponding to the first fundamental form (which is induced on the tangent spaces from the standard dot product onR3), and the above identifications are also consistent with these metrics. We now abstract these properties in a natural way in order to generalize the geometries from both chapters into one which subsumes them both, namely anabstract surfacecarrying aRiemannian metric.
Definition 8.3 An abstract (smooth) surface is a metric space S with homeomorphismsθi :Ui →Vifrom open subsetsUi ⊂S to open subsetsVi⊂R2 (iranging over an indexing setI) such that
(i) S =
i∈IUi, and
(ii) fori,j∈I, the mapφij :=θi◦θj−1:θj(Ui∩Uj)→θi(Ui∩Uj)is a diffeomorphism.
(iii) We shall assume also that the space isconnected.
As in the embedded case, we call theθicharts, the collection ofθianatlas, and the φijtransition functions(cf. Definition6.5). In what follows, we often just refer toS as a surface.
We say thatS isclosed if it is compact; e.g. by Bolzano–Weierstrass, an embedded surfaceS ⊂R3is compact if and only if it is closed inR3and bounded.
Given a continuous curveγ :[a,b] →S, we say thatγ issmoothif, whenever γ (t)∈Uifor some chartθi:Ui →Vi, the compositeθi◦γis locally a smooth curve onVi. Since, by definition, the transition functions are smooth, this condition does not depend on the choice of chart containingγ (t).
For an abstract smooth surfaceS (equipped with an atlas), aRiemannian metric onS is defined to be given byRiemannian metricson the images Vi of the charts θi:Ui→Vi⊂R2, subject to compatibility conditions that for alli,j(andφ=φij)
dφP(a),dφP(b)φ(P)= a,bP
inner-product given by metric onVi
inner-product given by metric onVj
for allP ∈θj(Ui∩Uj)anda,b∈ R2. This is just the statement that the transition functions are isometries of the Riemannian metrics on the open subsets ofR2, in the sense of Chapter4.
This enables us to define lengths and energies of curves on an abstract surfaceS, areas of regions onS, and geodesics onS merely by looking at the corresponding charts (this is entirely analogous to the case of embedded surfaces). The fact that these concepts are well defined follows from the invariance of length, energy and area under isometries; this was proved in Chapter4for lengths and areas, and the proof given there for lengths works equally well for energies.
Example The three classical geometries we studied before are:
(i) The Euclidean planeR2, with Riemannian metricdx2+dy2.
(ii) The embedded surfaceS2 ⊂ R3, with metric induced from the Euclidean metric onR3.
(iii) The unit disc model D of the hyperbolic plane with Riemannian metric 4(dx2+dy2)/(1−(x2+y2))2, or equivalently the upper half-plane modelH with Riemannian metric(dx2+dy2)/y2.
In the cases (i) and (iii), we only need one chart (θ =id) to define the abstract surface with its Riemannian metric. By a theorem of Hilbert, the hyperbolic plane cannot be realized as an embedded surface.
Example We saw before that the torusT ⊂ R3has natural chartsθ :U →V, arising from the projectionϕ :R2→T, whereU is the complement of two circles inT andV is the interior of a unit square inR2. In this case, the transition functions associated to an atlas consisting of such charts arelocallyjust translations inR2, and so clearly satisfy the required isometry condition, with respect to the Euclidean metric
8.2 ABSTRACT SMOOTH SURFACES AND ISOMETRIES 157 onR2— see however Exercise8.1. From this, it follows thatTis an abstract smooth surface, which comes equipped with the locally Euclidean Riemannian metric.
There are different Riemannian metrics which can be placed onT. One is the metric that arises from consideringTas an embedded surface inR3, and this we encountered in Chapter6(where we found it convenient to study the embedded surface as a surface of revolution). The more natural metric onTis however the locally Euclidean metric.
We observed in Chapter6thatT with this metric cannot be realized as an embedded surface inR3. AsTis compact, it is also not homeomorphic to an open subset ofR2. Given an abstract surfaceS equipped with a Riemannian metric, we can define an associated metric onSin the same way as Section4.3by
ρS(x1,x2)=inf{length:a piecewise smooth curve joiningx1andx2}.
The implications of this are twofold. The usual definition of an abstract surface is in terms of a Hausdorff topological space rather than a metric space; in the presence of a Riemannian metric however, there is a natural metric on the space defined by the above recipe. Moreover, if we have taken our definition of an abstract surface in terms of a metric space, then the given metric may well not be the natural metric to consider. For instance, for an embedded surfaceS ⊂R3, we have a metric onS given by taking distances between points inR3, but (assumingSto be connected) the natural metric to consider is the one induced from the Riemannian metric onS (as given by the first fundamental form).
Definition 8.4 A mapf : X → Y between abstract surfaces issmoothif for any chartsθ : U → V onX andθ∗ : U∗ → V∗ onY withU ∩f−1(U∗) = ∅, the composite map
f¯=θ∗◦f ◦θ−1:θ(U∩f−1(U∗))⊂V →V∗
is smooth. This is saying that, once we have identified the domains of charts onX andY as open subsets ofR2via the relevant charts, the induced map between the appropriate open subsets ofR2is smooth.
A smooth mapf is called adiffeomorphismif it has a smooth inverse.
Suppose nowX andY have Riemannian metrics. A smooth mapf is called alocal isometry if for all pairs of charts as above,f¯ preserves the Riemannian metric, that is, for allP∈θ(U∩f−1(U∗))anda,b∈R2,
a,bP= df¯P(a),df¯P(b)∗f¯(P). inner-product
forθchart
inner-product forθ∗chart
Iff is also a diffeomorphism, then it is called anisometry— lengths and areas are then preserved underf. Moreover, an isometry will also preserve the associated intrinsic metrics we defined above, i.e.ρY(f(x1),f(x2))=ρX(x1,x2)for allx1,x2∈X.
Example
(i) In Chapter5, we defined an isometry between the upper half-plane and the disc models of the hyperbolic planeH →D— in fact, we defined the Riemannian metric onH in order that our given map was an isometry.
(ii) We have the quotient map of surfacesϕ:R2→T, whereT is the torus.
Taking the locally Euclidean metric onT and the Euclidean metric onR2, we know thatϕis a local isometry, but it is clearly not an isometry.
(iii) On a compact orientable surfaceSof genusg(i.e. ag-holed torus), one can prove the existence of a locally hyperbolic Riemannian metric onS, and also that there exists a local isometryf :D→Sfrom the hyperbolic plane toS.
The theory from Chapter7 extends immediately to any abstract smooth surfaceS equipped with a Riemannian metric, to yield a normal neighbourhood W for any pointP ∈ S, and local geodesic polar coordinates(ρ,θ), with respect to which the metric takes the formdρ2+G(ρ,θ)dθ2; hereG(ρ,θ)is a smooth function in the coordinates. Theorem8.1suggests that we shoulddefine, for all points ofW \ {P}, theGaussian curvatureby the formula
K:= −(√ G)ρρ/√
G.
By Theorem8.1, this is consistent with our definition of the Gaussian curvature in the embedded case. It will prove to be slightly more convenient to impose a further condition on the normal neighbourhoodsWwe use, namely that they should bestrong normal neighbourhoods.
Definition 8.5 A normal neighbourhoodWfor which any two points are connected by at most one geodesic inW will be called astrong normal neighbourhood. For instance, on the sphere, open balls of radiusδ >0 are normal neighbourhoods when δ < π, but are not strong normal neighbourhoods unlessδ < π/2. More generally, we shall see in Proposition8.12that sufficiently small normal neighbourhoods of a point are always strong normal neighbourhoods (implied by the definition of strongly convex in Section8.4). The reader should perhaps be warned that the terminology introduced here is non-standard.
For the above to be a good definition of curvature, we need to prove two things:
(i) For anyQ∈S, we can find a strong normal neighbourhoodW of some pointP=Q withQ∈W.
(ii) The value ofKdoes not depend on the particular choice of pointPand strong normal neighbourhoodW.