Differential Geometry

Definition 19

Definition 20

Definition 21

Definition 22

Therefore, a vector field can be thought of as a curve through the tangent bundle of a manifold.

Definition 23

A Lie Derivative is essentially a directional derivative, and it measures how a function changes along a vector field.

Definition 24

It is helpful when chaining Lie Brackets since we can denote

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

We can also use relate repeated Lie Derivatives to doing repeated Lie Brackets.

Theorem 5

Definition 25

Definition 26

Distributions have different properties which are important to look at.

Definition 27

Definition 28

In involutive distributions, you can never leave the distribution by traveling along vectors inside the distribution.

Definition 29

It turns out that integrability and involutivity are equivalent to each other.

Theorem 6 (Frobenius Theorem)

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