# Differential Geometry

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#### Definition 19

$$M\subset \mathbb{R}^n$$ is a $$m$$-dimensional smooth sub-manifold of $$\mathbb{R}^n$$ if $$\forall \boldsymbol{p}\in M,\ \exists r > 0$$ and $$F: B_r(\boldsymbol{p}) \to \mathbb{R}^{n-m}$$ such that

$$\begin{aligned} M \cap B_r(\boldsymbol{p}) = \{\boldsymbol{x}\in B_r(\boldsymbol{p}) | F(\boldsymbol{x}) = 0\},\\ F\text{ is smooth,}\\ \forall \bar{\boldsymbol{x}} \in M \cap B_r(\boldsymbol{p}), \text{Rank}\left(\frac{\partial F}{\partial \boldsymbol{x}} \bigg\rvert_{\bar{\boldsymbol{x}}}\right) = n - m \end{aligned}$$
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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.

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#### Definition 20

The tangent space of a manifold $$M$$ at $$\boldsymbol{p}\in M$$ is given by

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

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#### Definition 21

The Tangent Bundle of a manifold $$M$$ is the collection of all tangent spaces

$$T_M = \bigcup_{\boldsymbol{p}\in M} T_{\boldsymbol{p}} M$$
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#### Definition 22

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

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#### Definition 23

The Lie Derivative of a function $$V$$ with respect to a vector field $$f$$ is given by

$$L_fV = (\nabla_{\boldsymbol{x}}V)^\top f(\boldsymbol{x}).$$
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A Lie Derivative is essentially a directional derivative, and it measures how a function changes along a vector field.

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#### Definition 24

Suppose that $$f(\boldsymbol{x})$$ and $$g(\boldsymbol{x})$$ are vector fields. The Lie Bracket of $$f$$ and $$g$$ is given by

$$[f, g] = L_fg - L_gf$$
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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

$$\text{ad}_fg = [f, g].$$

It is helpful when chaining Lie Brackets since we can denote

$$[f,[f,[f,\cdots[f,g]]]] = \text{ad}_f^ig.$$

Since the Lie Bracket is a vector field, we can look at Lie Derivatives with respect to the Lie Bracket of two vector fields.

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#### Theorem 4

For a function $$h$$ and vector fields $$f$$ and $$g$$,

$$L_{[f,g]}h = L_fL_gh - L_gL_fh$$
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We can also use relate repeated Lie Derivatives to doing repeated Lie Brackets.

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#### Theorem 5

$$L_gL_f^ih(\boldsymbol{x}) = 0 \Leftrightarrow L_{\text{ad}_f^ig}h(\boldsymbol{x}) = 0$$
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#### Definition 25

Suppose $$f_1,f_2,\cdots,f_n$$ are vector fields. A distribution $$\Delta$$ is the span of the vector fields at each point $$\boldsymbol{x}$$:

$$\Delta(\boldsymbol{x}) = \text{span}\{f_1(\boldsymbol{x}), f_2(\boldsymbol{x}),\cdots,f_n(\boldsymbol{x})\}.$$
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At each point $$\boldsymbol{x},\ \Delta(\boldsymbol{x})$$ is a subspace of the tangent space at $$\boldsymbol{x}$$.

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#### Definition 26

The dimension of a distribution at a point $$\boldsymbol{x}$$ is given by

$$\text{Dim }\Delta(\boldsymbol{x}) = \text{Rank}\left(\begin{bmatrix} f_1(\boldsymbol{x}) & \bigg\lvert & f_2(\boldsymbol{x}) & \bigg\lvert & \cdots & \bigg\lvert & f_n(\boldsymbol{x}) \end{bmatrix}\right)$$
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Distributions have different properties which are important to look at.

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#### Definition 27

A distribution $$\Delta$$is nonsingular, also known as regular, if its dimension is constant.
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#### Definition 28

A distribution $$\Delta$$ is involutive if

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

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#### Definition 29

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

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#### Theorem 6 (Frobenius Theorem) <a href="#theorem-6" id="theorem-6"></a>

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