2. Smooth manifolds: A review

7

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]

p

such

that

[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