Before formalizing the concept of a Riemannian metric, we should recall various facts from Analysis about differentiating functions in several variables. This section 75
provides such revision as might be needed, but also allows us to fix on the most convenient notation for derivatives, at least for our purposes. The right choice of notation enables us for instance to give a precise meaning to the idea ofdifferentials, a concept treated rather cavalierly in some undergraduate textbooks, especially in Applied Mathematics.
SupposeUis an open subset ofRn; a mapf :U →Rmis defined coordinatewise by real-valued functions(f1,. . .,fm)onU. The mapf is calledsmooth(orC∞) if each fi has partial derivatives of all orders. A smooth map is certainly differentiable (for which having continuous partial derivatives suffices). Thederivativeoff ata ∈ U is a linear mapdfa:Rn→Rm(in some books, denotedDfaorf(a)) such that, for h=0,
f(a+h)−f(a)−dfa(h)
h →0 as h→0∈Rn.
Whenm=1, the linear mapdfa :Rn→Ris determined by the partial derivatives off ata, namely
∂
f
∂x1(a),. . .,∂∂xf
n(a)
via matrix multiplication, i.e.
(h1,. . .,hn)→
i
∂f
∂xi(a)hi.
In general, whenmis arbitrary,dfa:Rn→Rmis determined by them×nmatrix of partial derivatives ata, theJacobianmatrix
J(f)= ∂fi
∂xj
.
Example We consider analytic functionsf :U → Cin one complex variablez, whereU is an open subset ofC. By definition, this means that, for anyz∈U,
f(z+w)−f(z)−wf(z)
|w| →0 as 0=w→0∈C,
wherefdenotes the (complex valued) derivativedf/dz. So if, forP ∈ U, we set f(P)=a+ib, andw=h1+ih2, then
wf(P)=(ah1−bh2)+i(bh1+ah2).
If we now considerf as a mapU → R2, then the linear mapdfP : R2 → R2is represented by the matrix
a −b
b a .
Given any smooth real-valued functionsu(x,y), v(x,y)on an open set U ⊂ R2, and writingf(x+iy) = u(x,y)+iv(x,y), recall thatf is an analytic function of
4.1 REVISION ON DERIVATIVES AND THE CHAIN RULE 77 z =x+iyif and only if theCauchy–Riemannequations are satisfied, namely that
∂u/∂x = ∂v/∂y and∂u/∂y = −∂v/∂x hold everywhere. This essentially is the content of the above calculation.
In particular, we note that iff is a complex analytic function onU ⊂ C, andP a point ofU withf(P)non-zero, thendfP preserves angles and orientations, since all non-zero vectors are rotated through the same angle, namely the argument of the complex numberf(P). If two smooth curvesγ1,γ2pass though the pointP ∈ U, the angle at which they meet is defined to be the angle between the derivatives γ1 andγ2atP. If the two curves meet atPwith an angleα, then use of the Chain Rule shows that their imagesf ◦γ1andf ◦γ2meet atf(P)with the same angle and the same orientation.
Playing a central role in the rest of the book will be the Chain Rule. SupposeU ⊂Rn andV ⊂Rpare open subsets. Given smooth mapsf : U → Rm andg :V →U, then the compositefg:V →Rmis a smooth map, and has derivative atP∈V
d(fg)P =dfg(P)◦dgP.
In other words, this is summed up by the slogan that the derivative of the composite is just the composite of the derivatives. In terms of Jacobian matrices, it says
J(fg)P =J(f)g(P)J(g)P, where the multiplication here is just matrix multiplication.
Of particular importance will be the case whenU is an open subset ofRn and f :U →Ris smooth. For eachP∈U, we have the linear mapdfP :Rn→R(an element therefore of thedualspace toRn). These then yield a smooth map
df :U →Hom(Rn,R),
where Hom(Rn,R) denotes the dual vector space, consisting of homogeneous linear forms on Rn, where the dual space may also be identified with Rn. More generally, any smoothly varying family of homogeneous linear forms on Rn, parametrized byU(i.e. given by a smooth mapg :U →Hom(Rn,R)), is called a differentialonU.
To understand this concept concretely, we should work in coordinates; if the standard basis of Rn is denoted by e1,. . .,en, there are corresponding coordinate functionsxi :Rn → R, defined by projection onto theith coordinate. As these are already homogeneous linear forms on Rn, the derivative (dxi)P is the same linear form, independent ofP. Thus the differentialdxiis a constant function ofP, and may therefore be regarded as a fixed linear form, namely the corresponding coordinate function onRn, given by
dxi(a1,. . .,an) = ai.
The homogeneous linear formsdx1,. . .,dxn provide the dual basis to the standard basis e1,. . .,en of Rn, that is dxi(ej) = δij for all 1 ≤ i,j ≤ n. We may wish to change the origin inRn, and then the functionsxi may acquire constant terms;
however, the corresponding linear formsdxiare still just the elements of the standard dual basis. A general differential onU may therefore, by definition, be written in the formn
i=1gidxi, with thegismooth functions onU, or equivalently as the row vector(g1,. . .,gn)of smooth functions.
If nowf : U → Ris an arbitrary smooth function on an open setU ⊂Rn, the linear formdfP :Rn→Ris represented by the partial derivatives atP,
∂f
∂x1
(P),. . ., ∂f
∂xn
(P) ,
or equivalently
dfP=
i
∂f
∂xi(P)dxi.
Thus, as differentials, we have the familar identity that
df =
i
(∂f/∂xi)dxi.
The Chain Rule also translates into a familiar identity on differentials. If g = (g1,. . .,gn):U → Rn, whereU is an open subset ofRm(coordinatesu1,. . .,um
say) and thegiare smooth functions onU, then for any coordinate functionxionRn, we have
dxi= m
j=1
(∂gi/∂uj)duj,
as differentials onU. We shall use this in particular whenm=n, andg represents the map corresponding to a change of coordinatesxi =gi(u1,. . .,un). We often then write
dxi = n j=1
(∂xi/∂uj)duj.
Themoraltherefore of the last few paragraphs is that one may continue manipulating differentials in the familiar formal way, but we now have a rigorous interpretation as to what a differential is: namely, it is a smooth family of (homogeneous) linear forms.
There is a notational convention which we shall use frequently below. Ifα:Rn→ Randβ :Rn→Rare two linear (homogeneous) forms onRn, we have an obvious