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**Example text**

Let Z be a smooth vector field on N × R with R∂ component ∂t . 3 Isotopies 57 (1) Suppose Z is globally integrable. Then its flow Φ satisfies Φt (N × {s}) ⊂ N × {s + t}, s, t ∈ R and D : N × R → N, (x, t) → pr1 ◦Φt (x, 0) = Dt (x) is a diffeotopy of N . (2) If Z has in addition in the complement of the compact set C = K × [c, d] the N -component zero, then Z is globally integrable and D is constant in the complement of K. Proof. (1) Let α : R → N × R be the integral curve through (y, s). Then ∂ β = pr2 ◦α is the integral curve of ∂t through s, and therefore the relation β(t) = s + t holds.

This orientation of A is called the pre-image orientation. P 2 4. Let f : Rn → R, (xi ) → xi and S n−1 = f −1 (1). Then the pre-image orientation coincides with the boundary orientation with respect to S n−1 ⊂ Dn . 11 Tangent Bundle. Normal Bundle The notions and concepts of bundle theory can now be adapted to the smooth category. A smooth bundle p : E → B has a smooth bundle projection p and the bundle charts are assumed to be smooth. A smooth subbundle of a smooth vector bundle has to be defined by smooth bundle charts.

Proof. The map ϕv is an immersion if for each x ∈ M the kernel of Tx pv has trivial intersection with Tx M . The differential of pv is again pv . Hence 0 = z = pv (z) + λv ∈ Tx M is contained in the kernel of Tx pv if and only if z = λv and hence v is a unit vector in Tx M . 5) Theorem. Let M be smooth compact n-manifold. Let f : M → R2n+1 be a smooth map which is an embedding on a neighbourhood of a compact subset A ⊂ M . Then there exists for each ε > 0 an embedding g : M → R2n+1 which coincides on A with f and satisfies f (x) − g(x) < ε for x ∈ M .