Vector Fields and Lie Algebra

Vector Fields on Manifolds

Given a differentiable manifold , a vector field on is a section of the tangent bundle. Explicitly, is an assignment that sends each point to a tangent vector . The space of all vector fields is denoted by . or .

e.g. .

Vector Fields are Derivations

One can see from the derivation definition of tangent vectors that a vector field can be identified as a derivation that satisfies the equation:

The space of vector fields has a rich structure:

Proposition

The space of vector fields is a -vector space.

Proof It suffices to check that the scaler multiplication is valid. Indeed, where the dot denotes the scalar multiplication in . It is clear that this satisfies distributivity and the associativity.

Proposition

The space of vector fields is a module over the ring .

Proof We can define the the scaler multiplication of a vector field by a smooth function as follows: where the dot denotes the scalar multiplication in . Clearly this gives a smooth vector field, and thus makes a module over .

Pullback Vector Fields

Pullback Vector Field

Suppose is a local diffeomorphism. The pullback of a vector field by is the pullback section of the tangent bundle. That is

Proof

Lie Bracket of Vector Fields

Lie Bracket of Vector Fields

Let . Then the Lie bracket of and is the vector field defined (as a derivation) by
The Lie bracket is thus the commutator of and as operators on smooth functions.

Properties of Lie Bracket

The following hold for the Lie bracket of vector fields:

1. Bilinearity: .
2. Skew-symmetry: .
3. Jacobi Identity: .

That is, forms a real Lie algebra.

Proof Bilinearity and skew-symmetry follow from the definition directly. We now check Jacobi identity. For all , there holds

Corollary

Let be a smooth manifold and let be an immersed submanifold with or without boundary in . If and are smooth vector fields on that are tangent to , then is also tangent to .