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Differential Geometry

Definition 19

MRnM\subset \mathbb{R}^n
is a
mm
-dimensional smooth sub-manifold of
Rn\mathbb{R}^n
if
pM, r>0\forall \boldsymbol{p}\in M,\ \exists r > 0
and
F:Br(p)RnmF: B_r(\boldsymbol{p}) \to \mathbb{R}^{n-m}
such that
MBr(p)={xBr(p)F(x)=0},F is smooth,xˉMBr(p),Rank(Fxxˉ)=nm\begin{aligned} M \cap B_r(\boldsymbol{p}) = \{\boldsymbol{x}\in B_r(\boldsymbol{p}) | F(\boldsymbol{x}) = 0\},\\ F\text{ is smooth,}\\ \forall \bar{\boldsymbol{x}} \in M \cap B_r(\boldsymbol{p}), \text{Rank}\left(\frac{\partial F}{\partial \boldsymbol{x}} \bigg\rvert_{\bar{\boldsymbol{x}}}\right) = n - m \end{aligned}
By Definition 19, a manifold is essentially defined as the 0-level set of some smooth function
FF
and can be thought of as a surface embedded in a higher dimension.

Definition 20

The tangent space of a manifold
MM
at
pM\boldsymbol{p}\in M
is given by
TpM=Null(Fxp)T_{\boldsymbol{p}}M = \text{Null}\left(\frac{\partial F}{\partial \boldsymbol{x}}\bigg |_{\boldsymbol{p}}\right)
The tangent space consists of all vectors tangent to the manifold at a particular point
p\boldsymbol{p}
.

Definition 21

The Tangent Bundle of a manifold
MM
is the collection of all tangent spaces
TM=pMTpMT_M = \bigcup_{\boldsymbol{p}\in M} T_{\boldsymbol{p}} M

Definition 22

A vector field
f:MTMf:M\to T_M
on a manifold
MM
is an assignment of each point
pM\boldsymbol{p}\in M
to a vector in the tangent space in that point
TpMT_{\boldsymbol{p}}M
.
Therefore, a vector field can be thought of as a curve through the tangent bundle of a manifold.

Definition 23

The Lie Derivative of a function
VV
with respect to a vector field
ff
is given by
LfV=(xV)f(x).L_fV = (\nabla_{\boldsymbol{x}}V)^\top f(\boldsymbol{x}).
A Lie Derivative is essentially a directional derivative, and it measures how a function changes along a vector field.

Definition 24

Suppose that
f(x)f(\boldsymbol{x})
and
g(x)g(\boldsymbol{x})
are vector fields. The Lie Bracket of
ff
and
gg
is given by
[f,g]=LfgLgf[f, g] = L_fg - L_gf
The Lie Bracket is another vector field, and it essentially measures the difference between moving along vector field
ff
and vector field
gg
across some infinitesimal distance. Another way to think about the Lie Bracket is as a measure of the extent to which
ff
and
gg
commute with each other. The Lie Bracket is also sometimes denoted using the adjoint map
adfg=[f,g].\text{ad}_fg = [f, g].
It is helpful when chaining Lie Brackets since we can denote
[f,[f,[f,[f,g]]]]=adfig.[f,[f,[f,\cdots[f,g]]]] = \text{ad}_f^ig.
Since the Lie Bracket is a vector field, we can look at Lie Derivatives with respect to the Lie Bracket of two vector fields.

Theorem 4

For a function
hh
and vector fields
ff
and
gg
,
L[f,g]h=LfLghLgLfhL_{[f,g]}h = L_fL_gh - L_gL_fh
We can also use relate repeated Lie Derivatives to doing repeated Lie Brackets.

Theorem 5

LgLfih(x)=0Ladfigh(x)=0L_gL_f^ih(\boldsymbol{x}) = 0 \Leftrightarrow L_{\text{ad}_f^ig}h(\boldsymbol{x}) = 0

Definition 25

Suppose
f1,f2,,fnf_1,f_2,\cdots,f_n
are vector fields. A distribution
Δ\Delta
is the span of the vector fields at each point
x\boldsymbol{x}
:
Δ(x)=span{f1(x),f2(x),,fn(x)}.\Delta(\boldsymbol{x}) = \text{span}\{f_1(\boldsymbol{x}), f_2(\boldsymbol{x}),\cdots,f_n(\boldsymbol{x})\}.
At each point
x, Δ(x)\boldsymbol{x},\ \Delta(\boldsymbol{x})
is a subspace of the tangent space at
x\boldsymbol{x}
.

Definition 26

The dimension of a distribution at a point
x\boldsymbol{x}
is given by
Dim Δ(x)=Rank([f1(x)f2(x)fn(x)])\text{Dim }\Delta(\boldsymbol{x}) = \text{Rank}\left(\begin{bmatrix} f_1(\boldsymbol{x}) & \bigg\lvert & f_2(\boldsymbol{x}) & \bigg\lvert & \cdots & \bigg\lvert & f_n(\boldsymbol{x}) \end{bmatrix}\right)
Distributions have different properties which are important to look at.

Definition 27

A distribution
Δ\Delta
is nonsingular, also known as regular, if its dimension is constant.

Definition 28

A distribution
Δ\Delta
is involutive if
f,gΔ,[f,g]Δ\forall f, g\in \Delta, \quad [f, g] \in \Delta
In involutive distributions, you can never leave the distribution by traveling along vectors inside the distribution.

Definition 29

A nonsingular
KK
-dimensional distribution
Δ(x)=span{f1(x),,fk(x)}\Delta(\boldsymbol{x}) = \text{span}\{f_1(\boldsymbol{x}), \cdots, f_k(\boldsymbol{x})\}
is completely integrable if
ϕ1,,ϕnk\exists \phi_1,\cdots,\phi_{n-k}
such that
i,k, Lfkϕi=0\forall i,k,\ L_{f_k}\phi_i = 0
and
ablaxϕiabla_{\boldsymbol{x}}\phi_i
are linearly independent. \label{thm:involutive}
It turns out that integrability and involutivity are equivalent to each other.

Theorem 6 (Frobenius Theorem)

A nonsingular
Δ\Delta
is completely integrable if and only if
Δ\Delta
is involutive.