Therank of a linear transformationL:V −→W between finite-dimensional vector spaces is the dimension of the imageL(V )as a subspace ofW. IfLis represented by a matrixArelative to a basis forV and a basis forW, then the rank ofLis the same as the rank ofA, because the imageL(V )is simply the column space ofA.
Now consider a smooth mapf: N −→ M of manifolds. Its rank at a point p in N, denoted by rkf (p), is defined as the rank of the differential f∗,p : Tp(N ) −→ Tf (p)(M). Relative to the coordinate neighborhoods (U, x1, . . . , xn) at p and(V , y1, . . . , ym)atf (p), the differential is represented by the Jacobian matrix[∂fi/∂xj(p)](Proposition 8.12), so
rkf (p)=rk ∂fi
∂xj(p)
.
Since the differential of a map is independent of the coordinate chart, so is its rank.
Definition 8.19.A pointpinNis acritical pointoff if the differential f∗,p:TpN −→Tf (p)M
fails to be surjective. It is aregular pointoff if the differentialf∗,pis surjective.
A point inMis acritical valueif it is the image of a critical point; otherwise it is a regular value.
×
×
×
×
×
=critical points
=critical values f
N
M
Fig. 8.4.Critical points and critical values.
Two aspects of this definition merit elaboration:
8.8 Rank, Critical and Regular Points 87 (i) We donot define a regular value to be the image of a regular point. In fact, a regular value need not be in the image off at all. Any point ofMnot in the image off is automatically a regular value because it is not the image of a critical point.
(ii) A point c inM is a critical value if and only if somepoint in the preimage f−1({c})is a critical point. A pointcin the image off is a regular value if and only ifeverypoint in the preimagef−1({c})is a regular point.
Proposition 8.20.For a real-valued functionf: M−→R, a pointpinMis a critical point if and only if relative to some chart(U, x1, . . . , xn)containingp, all the partial derivatives
∂f
∂xj(p)=0, j =1, . . . , n.
Proof. By Proposition 8.12 the differentialf∗,p: TpM −→ Tf (p)R Ris repre- sented by the matrix
∂f
∂x1(p) . . . ∂f
∂xn(p)
.
Since the image off∗,p is a linear subspace ofR, it is either zero-dimensional or one-dimensional. In other words,f∗,p is either the zero map or a surjective map.
Therefore,f∗,pfails to be surjective if and only if all the partial derivatives∂f/∂xi(p) are zero.
Problems
8.1.* Differential of a map LetF:R2−→R3be the map
(u, v, w)=F (x, y)=(x, y, xy).
ComputeF∗(∂/∂x)as a linear combination of∂/∂u,∂/∂v, and∂/∂w.
8.2. Differential of a map
Fix a real numberαand defineF:R2−→R2by u
v
=(u, v)=F (x, y)=
cosα −sinα sinα cosα
x y
.
LetX = −y ∂/∂x+x ∂/∂ybe a vector field onR2. IfF∗(X)=a ∂/∂u+b ∂/∂v, findaandbin terms ofx,y, andα.
8.3. Transition matrix for coordinate vectors
Letx, ybe the standard coordinates onR2, and letUbe the open set U=R2− {(x,0)|x≥0}.
OnUthe polar coordinatesr, θ are uniquely defined by
x=rcosθ,
y=rsinθ, r >0, 0< θ <2π.
Find∂/∂rand∂/∂θin terms of∂/∂xand∂/∂y.
8.4.* Velocity of a curve in local coordinates Prove Proposition 8.15.
8.5. Velocity vector
Letp=(x, y)be a point inR2. Then cp(t )=
cos 2t −sin 2t sin 2t cos 2t
x y
, t ∈R,
is a curve with initial pointpinR2. Compute the velocity vectorcp(0).
8.6. Differential of a linear map
LetL:Rn −→Rmbe a linear map. For anyp∈Rn, there is a canonical identification:
Tp(Rn)−→∼ Rngiven by
ai ∂
∂xi →a= a1, . . . , an.
Show that the differential L∗,p: Tp(Rn) −→ Tf (p)(Rm)is the mapL:Rn −→ Rm itself, with the identification of the tangent spaces as above.
8.7.* Tangent space to a product
IfMandNare manifolds, letπ1:M×N −→Mandπ2:M×N −→N be the two projections. Prove that for(p, q)∈M×N,
π1∗×π2∗:T(p,q)(M×N )−→TpM×TqN is an isomorphism.
8.8. Differentials of multiplication and inverse
LetGbe a Lie group with multiplication mapà: GìG−→ G, inverse mapι:G
−→G, and identity elemente.
(a) Show that the differential at the identity of the multiplication mapàis addition:
à∗,(e,e):TeGìTeG−→TeG, à∗,(e,e)(Xe, Ye)=Xe+Ye.
(Hint: First, computeà∗,(e,e)(Xe,0)andà∗,(e,e)(0, Ye)using Proposition 8.17.) (b) Show that the differential at the identity ofιis the negative:
ι∗,e:TeG−→TeG, ι∗,e(Xe)= −Xe. (Hint: Take the differential ofà(c(t ), (ι◦c)(t ))=e.)
8.8 Rank, Critical and Regular Points 89 8.9.* Transforming vectors to coordinate vectors
LetX1, . . . , Xnbenvector fields on an open subsetUof a manifold of dimension n. Suppose that atp ∈ U, the vectors(X1)p, . . . , (Xn)p are linearly independent.
Show that there is a chart(V , x1, . . . , xn)aboutpsuch that(Xi)p =(∂/∂xi)pfor alli=1, . . . , n.
9
Submanifolds
We now have two ways of showing that a given topological space is a manifold:
(a) by checking directly that the space is Hausdorff, second countable, and has aC∞ atlas;
(b) by exhibiting it as an appropriate quotient space. Chapter 7 lists some conditions under which a quotient space is a manifold.
In this chapter we introduce the concept of aregular submanifoldof a manifold, a subset that is locally defined by the vanishing of some of the coordinate functions.
Using the inverse function theorem, we derive a criterion, called theregular level set theorem, that can often be used to show that a level set of aC∞map of manifolds is a regular submanifold and therefore a manifold.
Although the regular level set theorem is a simple consequence of the constant rank theorem to be discussed in Chapter 11, deducing it directly from the inverse function theorem has the advantage of producing explicit coordinate functions on the submanifold.