MAT 531 (Spring 1996) Topology/Geometry II

Week 11
It follows from Theorem 2 of Week 10 and a standard compact --> open argument that (keeping the same context as that theorem) there exists a positive epsilon such that alpha[(-epsilon,epsilon) x B_{a/2}(x_0)] is contained in B_a(x_0). So for |t| < epsilon, and x in B_{a/2}(x_0), the point alpha(t,x) is an admissible initial condition for the problem considered in Theorem 1, i.e. for |s| < b, alpha(s, alpha(t,x)) is a point of U.

Theorem 3.(See Spivak, pp. 201,202). For x in B_{a/2}(x_0), |t| < epsilon, and |s| and |s+t| < b,
alpha(s, alpha(t,x)) = alpha(s+t,x).

It follows that for |t| < epsilon the map phi_t defined by phi_t (x) = alpha(t,x) is a homeomorphism (in fact, a diffeomorphism when we can suppose that f is of class C^{infty}) of B_{a/2}(x_0) onto its image, with inverse phi_{-t}. Because when s and t are sufficiently small we have
phi_s o phi_t = phi_{s+t}
the ``group'' of these diffeomorphisms has the structure of the additive group of real numbers in a neighborhood of 0. The phi's can be called a local one-parameter group of local diffeomorphisms.

Class exercise. Show that the ``law of exponents'' (e^s)(e^t) = e^{s+t} is a special case of this phenomenon.

All this analysis can be moved onto a manifold:

Theorem 4. (Spivak, Theorem 5.5) Let X be a smooth vectorfield on a manifold M, and p a point of M. Then there exists a neighborhood V of p, an epsilon > 0 and and a local one-parameter group of local diffeomorphisms phi_t : V --> M (|t| < epsilon) satisfying

phi_s o phi_t = phi_{s+t} (for |t|,|s|,|s+t| < epsilon)
(d/dt)[phi_t(x)] = X(phi_t(x)).

The two occurrences of ``local'' can be deleted in case the vectorfield X has compact support (in particular, when M is compact).

Theorem 5. (Spivak, Theorem 5.6) Let X be a smooth vectorfield, with compact support, on a manifold M. Then there exists one-parameter group of diffeomorphisms phi_t : M --> M satisfying

phi_s o phi_t = phi_{s+t}
(d/dt)[phi_t(x)] = X(phi_t(x)).

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