Here are two sample theorems that we are going to prove in these
lectures. To state the first one, we introduce some standard notation.
Given manifolds M and TV, we write
N) for the set of r times
continuously differentiable maps from M to N (resp. continuous maps if
r = 0). This presupposes, of course, that M and N are at least of class
Cr. We use 'smooth' synonymously with C°°. The tangent bundle
of a differentiable manifold M is denoted by TM. A Cr-immersion
f: M -• T V is an / G
TV), r 1, whose differential Tf: TM -•
T7V is fibrewise injective. Recall that a manifold M is called closed if
each component of M is compact and without boundary; it is called
open if each component is noncompact or with nonempty boundary.
Usually we shall implicitly assume that our manifolds are connected; in
that case 'open5 and 'closed' are complementary notions. The following
theorem is commonly known as the Smale-Hirsch theorem. It was
proved in the special case M = Sm, T V = IRg, g m + l, byS. Smale,
and in its general form by M. Hirsch. There are other proofs due to
R. Palais and V. Poenaru.
1.1 (Smale-Hirsch). Let M and N be smooth manifolds.
Assume that dimM dim iV or that M is open. Then f G C°(M, N)
is homotopic to a smooth immersion of M into N if and only if f is
covered by a fibrewise injective bundle map F G C®(TM,TN).
To say that / is covered by F means that we have a commutative
TM —F-^ TN
M —^- TV,
where the vertical maps are the obvious projections.