Differential Geometry
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By Definition 19, a manifold is essentially defined as the 0-level set of some smooth function and can be thought of as a surface embedded in a higher dimension.
The tangent space consists of all vectors tangent to the manifold at a particular point .
A vector field on a manifold is an assignment of each point to a vector in the tangent space in that point .
Therefore, a vector field can be thought of as a curve through the tangent bundle of a manifold.
A Lie Derivative is essentially a directional derivative, and it measures how a function changes along a vector field.
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.
We can also use relate repeated Lie Derivatives to doing repeated Lie Brackets.
Distributions have different properties which are important to look at.
In involutive distributions, you can never leave the distribution by traveling along vectors inside the distribution.
It turns out that integrability and involutivity are equivalent to each other.
Suppose that and are vector fields. The Lie Bracket of and is given by
The Lie Bracket is another vector field, and it essentially measures the difference between moving along vector field and vector field across some infinitesimal distance. Another way to think about the Lie Bracket is as a measure of the extent to which and commute with each other. The Lie Bracket is also sometimes denoted using the adjoint map
For a function and vector fields and ,
Suppose are vector fields. A distribution is the span of the vector fields at each point :
At each point is a subspace of the tangent space at .
The dimension of a distribution at a point is given by
A distribution is nonsingular, also known as regular, if its dimension is constant.
A distribution is involutive if
A nonsingular -dimensional distribution is completely integrable if such that and are linearly independent. \label{thm:involutive}
A nonsingular is completely integrable if and only if is involutive.