M⊂Rn is a m-dimensional smooth sub-manifold of Rn if ∀p∈M,∃r>0 and F:Br(p)→Rn−m such that
M∩Br(p)={x∈Br(p)∣F(x)=0},F is smooth,∀xˉ∈M∩Br(p),Rank(∂x∂Fxˉ)=n−m
By Definition 19, a manifold is essentially defined as the 0-level set of some smooth function F and can be thought of as a surface embedded in a higher dimension.
The tangent space of a manifold M at p∈M is given by
TpM=Null(∂x∂Fp)
The tangent space consists of all vectors tangent to the manifold at a particular point p.
The Tangent Bundle of a manifold M is the collection of all tangent spaces
TM=⋃p∈MTpM
A vector field f:M→TM on a manifold M is an assignment of each point p∈M to a vector in the tangent space in that point TpM.
The Lie Derivative of a function V with respect to a vector field f is given by
LfV=(∇xV)⊤f(x).
Suppose that f(x) and g(x) are vector fields. The Lie Bracket of f and g is given by
[f,g]=Lfg−Lgf
The Lie Bracket is another vector field, and it essentially measures the difference between moving along vector field f and vector field g across some infinitesimal distance. Another way to think about the Lie Bracket is as a measure of the extent to which f and g commute with each other. The Lie Bracket is also sometimes denoted using the adjoint map
adfg=[f,g].
[f,[f,[f,⋯[f,g]]]]=adfig.
For a function h and vector fields f and g,
L[f,g]h=LfLgh−LgLfh
LgLfih(x)=0⇔Ladfigh(x)=0
Suppose f1,f2,⋯,fn are vector fields. A distribution Δ is the span of the vector fields at each point x:
Δ(x)=span{f1(x),f2(x),⋯,fn(x)}.
At each point x,Δ(x) is a subspace of the tangent space at x.
The dimension of a distribution at a point x is given by
Dim Δ(x)=Rank([f1(x)f2(x)⋯fn(x)])
A distribution Δis nonsingular, also known as regular, if its dimension is constant.
A distribution Δ is involutive if
∀f,g∈Δ,[f,g]∈Δ
A nonsingular K-dimensional distribution Δ(x)=span{f1(x),⋯,fk(x)} is completely integrable if ∃ϕ1,⋯,ϕn−k such that ∀i,k,Lfkϕi=0 and ablaxϕiare linearly independent. \label{thm:involutive}
A nonsingular Δ is completely integrable if and only if Δis involutive.