(αβ)(x,y):=(α(x)β(y)+α(y)β(x)) /2.
For real numbersλ1,λ2,à1,à2and linear formsα1,α2,β1,β2, we note the equality of bilinear forms
(λ1α1+λ2α2)(à1β1+à2β2)=λ1à1α1β1+λ1à2α1β2+λ2à1α2β1+λ2à2α2β2. Finally in this section, we should mention the Inverse Function theorem, which will be needed in some subsequent chapters. This states that, ifUis an open subset ofRnand f :U → Rnis a smooth map, with the Jacobian matrixJ(f)non-singular at some pointP ∈ U, then locallyf is a homeomorphism from some open neighbourhood V P (with V ⊂ U) onto an open subset V ⊂ Rn, and that the inverse map g :V→V is also smooth. We shall say thatf is locally adiffeomorphism.
Most references for the Inverse Function theorem (for instance Theorem 9.24 of [11]) prove it in the form that, iff is continuously differentiable, then so too is the local inverseg. However, the Chain Rule then givesJ(g)as the inverse ofJ(f), and so by Cramer’s Rule, we may express the partial derivatives∂g/∂yjas rational functions of the partial derivatives ∂f/∂xi of f. Thus, iff is alsosmooth(i.e. has partial derivatives of all orders), an inductive argument on the order shows that the same will be true forg.
4.2 Riemannian metrics on open subsets of R2
For notational simplicity, we now restrict ourselves to the casen=2. Suppose thatV is an open subset ofR2, and let the standard coordinates onR2be denoted by(u,v).
ARiemannian metriconV is defined by givingsmoothfunctionsE,F,GonV, such that the matrix
E(P) F(P) F(P) G(P)
is positive definite for allP∈V. Thus (forP∈V), this determines an inner-product , PonR2, where
e1,e1P=E(P) e1,e2P=F(P) e2,e2P=G(P),
withe1,e2denoting the standard basis forR2. A Riemannian metric should be thought of as a family of inner-products on thetangent spaces(all identified withR2).
The coordinate functionsu:V →R,v:V →Rgive rise to linear formsduand dv onR2, comprising the dual basis to the standard basis ofR2, and hence also to
bilinearformsdu2,du dvanddv2onR2, where
du2(h,k)=du(h)du(k) ←→
1 0 0 0 du dv(h,k)= 1
2(du(h)dv(k)+du(k)dv(h)) ←→
0 12
1
2 0
dv2(h,k)=dv(h)dv(k) ←→
0 0 0 1 . Here, the right-hand column indicates the 2×2 matrix defining the bilinear form.
Thus, the family of bilinear forms onR2determined (forP∈V) by the matrix of smooth functions
E F
F G ,
may be then written simply asEdu2+2Fdu dv+Gdv2. When the matrix is positive definite for all P ∈ V, this yields a smooth family of inner-products on R2, and this is precisely what we defined to be a Riemannian metricon V. Note that the Euclidean case (when the inner-product is the Euclidean inner-product for allP∈V) corresponds to the constant functionsE=G=1 andF=0.
Given such a Riemannian metric onV, we may use this family of inner-products to define the length ofsmoothorpiecewise smoothcurves. For a smooth curveγ, we have its derivative, denotedγorγ˙; we defineγat a pointγ (t)by means of the inner-product determined atγ (t)by the given Riemannian metric. We shall refer to γas thespeedofγ att.
Definition 4.1 ForV an open subset ofR2, equipped with a Riemannian metric Edu2+2Fdu dv+Gdv2, thelengthof a smooth curve
γ =(γ1,γ2):[a,b] →V is defined to be
b
a
(Eγ˙12+2Fγ˙1γ˙2+Gγ˙22)1/2dt.
Example Consider the geometry ofV =R2with the Riemannian metric 4(du2+dv2)
(1+u2+v2)2.
This is an example of a metric which isconformalto the Euclidean metric, in that the inner-product at each pointPis just a scaling of the Euclidean inner-product, the scaling being defined by a smooth function ofP— in the above example, the function is 4/(1+u2+v2)2.
4.2 RIEMANNIAN METRICS ON OPEN SUBSETS OF R2 81 Let π : S2\ {N} → R2 be the stereographic projection map. Given a point P ∈ S2\ {N}, we haveπ(P) ∈ R2, and the Riemannian metric defines an inner- product, π(P)onR2. The tangent space toS2atPis defined to consist of vectorsx such thatxã−→
OP=0. Note that, under this definition, the tangent space is considered as a real vector space with origin at P. This definition will be consistent with the general definition we give in Chapter6for the tangent space at a given point on any embedded surface inR3.
P
O
There is an inverse mapσ :R2→S2\ {N}, given by σ (u,v)=
2u/(1+u2+v2), 2v/(1+u2+v2),(u2+v2−1)/(1+u2+v2) . Considered as a map to R3, we know that σ is smooth. Let us consider the two partial derivatives ofσ atπ(P), namelyσu(π(P))=(dσ)π(P)(e1)andσv(π(P))= (dσ)π(P)(e2). On all ofR2, we have thatσ(u,v)ãσ (u,v)=1; differentiating with respect touandv, we deduce thatσãσu=0 andσãσv =0 at all points ofR2, and hence that bothσu(π(P))andσv(π(P))are in the tangent space ofS2atP. Withσ as defined above, we check that
σu= 2
(u2+v2+1)2
−u2+v2+1,−2uv,−2u
σv= 2
(u2+v2+1)2
−2uv,u2−v2+1,−2v .
We note that, when evaluated at any point ofR2, the vectorsσuandσvare non-zero and orthogonal, and therefore are linearly independent vectors in R3; thus for all P ∈ S2, the derivative(dσ )π(P)induces an isomorphism of vector spaces between R2and the tangent space toS2atP.
Givenx1,x2in the tangent space atP, we claim that x1ãx2= dπP(x1),dπP(x2)π(P).
SincedπP◦(dσ)π(P)is the identity onR2, this is equivalent to the statement that (dσ)π(P)u1ã(dσ)π(P)u1= u1,u2π(P), (4.1) for allu1,u2 ∈R2. These equations say that the geometry onR2induced from the Riemannian metric 4(du2+dv2)/(1+u2+v2)2corresponds to the standard spherical geometry onS2\ {N}. Thus for instance, one may check that the length of a semi- infinite ray starting at the origin is justπ, since this is the length of the corresponding longitude onS2.
To check that Equation (4.1) holds for allu1,u2∈R2, we need only check (using the bilinearity) that it holds for the standard basis vectorse1,e2, which therefore reduces the problem to showing that
σuãσu= 4
(u2+v2+1)2, σuãσv=0 and σvãσv= 4 (u2+v2+1)2, at all points ofR2. To check these is an elementary manipulation.
The basic theory of this example will be generalized in Chapter 6 to arbitrary embedded surfaces inR3.