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
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
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