We want to work with spaces that locally look like . Informally this is exaclty the concept that a manifold captures.

A -dimensional smooth submanifold of is a subset of with the following properties. For all there is an open neighbourhood with with a diffemorphism , such that

We call a chart.

For a subset of the following are equivalent

1. is a k-dimensional smooth submanifold of cartesian space.

2. "Locally M is the zero set of a regular height function."

   For all points in there is an open neighbourhood of and a smooth map such that , is surjective and .

3. "Locally M is the graph of a smooth function."

   For all points in there is a diffeomorphism permuting coordinates, and open neighbourhood of and a smooth map such that .

We can define the n-dimensional sphere as the set of vectors in with norm one. There are several ways to see that this defines a manifold.

1. Using charts. We can give charts by using stereographic projections. The stereographic projection is only well defined onto if we take out the pole. We can simply use two stereographic projections, one missing the northpole, and one missing the southpole.

2. As a local zero set of a regular height function. Consider the function . Notice that iff . Notice that the differential of is surjective on since it is given by . Its only zero is the origin, which does not lie in .

3. Locally as the graph of a smooth function. Consider the map given by . By restriciting the domain we obtain a smooth well defined map whose image is the upper hemisphere (without the equator). We can use to hit the lower hemisphere. Using coordinate permutations we can move a small-enough neighbourhood of any point into the upper or lower hemisphere.

TODO: Details

A smooth k-dimensional manifold atlas for a set is an arbitrary collection of bijections with the following properties.

1. The domains cover
2. for all , is open in ,

3. for all , and are open in and

   is a diffeomorphism.

We call the elements of the atlas charts.

Let be a smooth atlas of some set . We call a map compatible with if is again a smooth atlas of

Let and be atlasses for a set . We say that and are compatible if is again an atlas.

There is a partial order on atlases of a set given by inclusion. We say that an atlas is maximal if it is maximal with respect to this partial order.

TODO: show that this is welldefined.

Is every atlas contained in a maximal atlas?

There is a canonical topology on every set with a manifold atlas given as follows. The open sets are simply the domains of charts.

TODO: Show that this defines a topology.

TODO

TODO

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.

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 .

Smoothness is independent of the choice of atlas.

TODO: show this

TODO

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

where .

The tangent space naturally comes equipped with the structure of a vectorspace.

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.

TODO

TODO: with curves

TODO

Let be a smooth manifold embedded into euclidean space. We define the tangent bundle as

It comes equipped with a projection called the foot projection. the preimage of a point under the foot projection is called the fibre of of the tangent bundle. We call the zero-section

1. The tangent bundle is a a manifold of twice the dimension of M.
2. The tangent bundle is a smooth sub-vector-bundle of

TODO

We want to construct the analogue of the tangent bundle just for abstract manifolds this time.

Let be a k-dimensional smooth manifold with atlas . We define the tangent bundle as the dijoint union of all tangent spaces.

TODO: equip this with charts to make it into a manifold

Let be a smooth manifold. A vectorfield is a smooth map into the tangent bundle such that for all points in , or equivalently such that the footpoint projection after is the identity.

The smooth vectorfields on a given manifold form a vectorspace. We denote it by

TODO: Show that this is indeed a vectorspace.

TODO

TODO

Let be a manifold of dimension . We call the tangent bundle trivial if we find pointwise linearly independent vectorfields. I.e vectorfields such that for all is linearly independent.

TODO: for odd n.

TODO: Why did we talk about these???

Given a smooth vector field for some smooth manifold and an initial point in . Then an integral curve through is a curve with

1. "initial condition"
2. for all "solution of the OdE is determined by the vector field"

The flow of a smooth vectorfield on a smooth manifold is a map , with open in and

1. for all "differential equation"
2. for all "initial condition"

Notice that a flow consists of many integral curves.

TODO: latitude vector fields

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.

Let be a smooth n-dimensional manifold equipped with a smooth vector field . For a every point where the does not vanish, there is a local chart of around , such that the is given by a constant function in this chart.

Concretely such that for all , the first basis vecotor of the standard basis.

The zeros of a smooth vectorfield are called critical points.

TODO: Contruct vector fields on with attracting / repelling / mixed critival points.

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

Set where is defined, then

Let be open, and smooth. Assume that . Then there exists an open neighbourhood of such that and an , with a unique smooth map such that

1. for all "initial value"
2. for all and "solution of the OdE "
3. If for the solution can be found on with then for every we find a neighbourhood of such that the (unique) solution can be found on .
4. For each , there is a maximal open interval with on which one can define the (unique) solution of the initial value problem. By (3.) this defines a maximal open subset such that the intersection with is an interval containing on which the solution function can be defined.

For every compact smooth manifold embedded in euclidiean space and smooth vectorfield on , there is a unique flow of .

Let be a connected compact smooth manifold of positive dimension, equipped with a smooth function with exactly two critical points, then is homeomorphic, but not necessarily diffeomorphic, to the .

A smooth vectorfield on (or a suitable open subset) is called time-dependent if

for all . Here we use the cacnonical decomposition .

A flow of a time dependent smooth vectorfield is a smooth function satisfying

1. is open such that for all the set is an open interval containing .
2. "initial condition".
3. "time dependent differential equation".

TODO: Example of time dependent smooth vectorfield

Existence and uniqueness of flows of time dependent smooth vectorfields is just a special case of flow of general flow.

A subset of a smooth n-dimensional manifold is called a smooth submanifold of dimension if for all there is a smooth chart of with such that and . These charts of are called submanifold charts for

TODO: The equator sphere is a submanifold of and is homeomorphic to the .