2. Smooth manifolds: A review
Let X,Y e X(M). Define [X, Y] = XY - YX. This is a function
from F(M) to T(M) sending each / to X(Yf) - Y(Xf). An easy
computation shows that [X, Y] is a derivation on F(M), hence a
smooth vector field on M, which is called the bracket of X and Y.
The bracket assigns to each p e M the tangent vector [X, y]
[x,yy/) = xp(r/)-yp(.x-/).
Furthermore, the bracket operation has the following properties:
(a) [X, Y] = - [y, X] (skew-symmetry),
{b) [aX + &y, Z] = a[X, Z] + 6[y, Z],
[Z, aX + by] = a[Z, X] + 6[Z, y] (IR-bilinearity),
(c) [X, [y, Z]] + [y, [Z, X}] + [Z, [X, y]] - 0 (Jacobi identity).
The above properties say that the set X(M) with the operation
"bracket" of vector fields is a real Lie algebra. In general, a real
(respectively complex) Lie algebra is a real (respectively complex)
vector space V with an operation [ , ]: V x V V that satisfies
properties (a)-(c) above.
The bracket of vector fields has an interpretation as a derivation
of y along the "flows" of X to be explained now. The following propo-
sition is a manifold version of the existence and uniqueness theorem
for ordinary differential equations (see e.g. [Bo-Di, p. 37]).
Proposition 1.2. Let X be a smooth vector field on a smooth mani-
fold M, and let p £ M. Then there exists an open neighborhood U of
p, an open interval I around 0, and a smooth mapping fi : I xU M
such that the curve aq: I M given by aq(i) = /(t,q) (q G U) is
the unique curve that satisfies -^ = Xa ^ and aq(0) = q.
A curve with the above property is called an integral curve of
the vector field X. If £ is kept constant, the above proposition shows
that the assignment q \- aq (t) defines a function jt: U M on a
neighborhood U of p. This function is called the local flow of X. The
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