« Analysis on Manifolds

k-dimensional smooth manifold [009D]

A set with a smooth k-dimensional manifold atlas is called a manifold if the equipped with the canonical topology of the altas is hausdorff and second countable.

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Definition. smooth map between manifolds [009G]

Let be a map between smooth manifolds and . We say that is smooth if for all charts of and of the composition is a smooth map between subsets of .

Definition. Tangent space to a point of a manifold [009J]

For a smooth manifold with atlas we define the tangent space to the point in to be the following set of equivalence classes

where .

Defintion. Differential of a smooth map between manifolds [009L]

Let be a smooth map between manifolds. For a poing in and a chart around and a chart around we define the differential of at as

TODO: Show that this is well defined and independent of the choice of charts. TODO: Show the differential is functorial.

Theorem. Local existence and uniqueness of flows [00A3]

Let be a smooth smooth manifold equipped with a smooth vectorfield. For all point there is an open neighbourhood of and such that there is a flow of , uniquely determined by and the choice of domain.

There exists a maximal flow as well, defined on an open subset . Where is the maximal open interval containing 0, on which the flow line through can be defined.

Theorem. Flow Property [00A7]

Let be a smooth manifold equipped with a smooth vectorfield . Let be the maximal flow of .

Set where is defined, then