# Smooth manifolds (possibly with boundary or corners)

A smooth manifold is a manifold modelled on a normed vector space, or a subset like a
half-space (to get manifolds with boundaries) for which the change of coordinates are smooth maps.
We define a model with corners as a map `I : H → E`

embedding nicely the topological space `H`

in
the vector space `E`

(or more precisely as a structure containing all the relevant properties).
Given such a model with corners `I`

on `(E, H)`

, we define the groupoid of local
homeomorphisms of `H`

which are smooth when read in `E`

(for any regularity `n : with_top ℕ`

).
With this groupoid at hand and the general machinery of manifolds, we thus get the notion of `C^n`

manifold with respect to any model with corners `I`

on `(E, H)`

. We also introduce a specific type
class for `C^∞`

manifolds as these are the most commonly used.

## Main definitions

`model_with_corners 𝕜 E H`

: a structure containing informations on the way a space`H`

embeds in a model vector space E over the field`𝕜`

. This is all that is needed to define a smooth manifold with model space`H`

, and model vector space`E`

.`model_with_corners_self 𝕜 E`

: trivial model with corners structure on the space`E`

embedded in itself by the identity.`times_cont_diff_groupoid n I`

: when`I`

is a model with corners on`(𝕜, E, H)`

, this is the groupoid of local homeos of`H`

which are of class`C^n`

over the normed field`𝕜`

, when read in`E`

.`smooth_manifold_with_corners I M`

: a type class saying that the manifold`M`

, modelled on the space`H`

, has`C^∞`

changes of coordinates with respect to the model with corners`I`

on`(𝕜, E, H)`

. This type class is just a shortcut for`has_groupoid M (times_cont_diff_groupoid ⊤ I)`

.`ext_chart_at I x`

: in a smooth manifold with corners with the model`I`

on`(E, H)`

, the charts take values in`H`

, but often we may want to use their`E`

-valued version, obtained by composing the charts with`I`

. Since the target is in general not open, we can not register them as local homeomorphisms, but we register them as local equivs.`ext_chart_at I x`

is the canonical such local equiv around`x`

.

As specific examples of models with corners, we define (in the file `real_instances.lean`

)

`euclidean_space n`

for a model vector space of dimension`n`

.`model_with_corners ℝ (euclidean_space n) (euclidean_half_space n)`

for the model space used to define`n`

-dimensional real manifolds with boundary and`model_with_corners ℝ (euclidean_space n) (euclidean_quadrant n)`

for the model space used to define`n`

-dimensional real manifolds with corners

With these definitions at hand, to invoke an `n`

-dimensional real manifold without boundary,
one could use

```
variables {n : ℕ} {M : Type*} [topological_space M] [manifold (euclidean_space n)]
[smooth_manifold_with_corners (model_with_corners_self ℝ (euclidean_space n)) M]
```

.

However, this is not the recommended way: a theorem proved using this assumption would not apply
for instance to the tangent space of such a manifold, which is modelled on
`(euclidean_space n) × (euclidean_space n)`

and not on `euclidean_space (2 * n)`

! In the same way,
it would not apply to product manifolds, modelled on `(euclidean_space n) × (euclidean_space m)`

.
The right invocation does not focus on one specific construction, but on all constructions sharing
the right properties, like

```
variables {E : Type*} [normed_group E] [normed_space ℝ E] [finite_dimensional ℝ E]
{I : model_with_corners ℝ E E} [I.boundaryless]
{M : Type*} [topological_space M] [manifold E M] [smooth_manifold_with_corners I M]
```

Here, `I.boundaryless`

is a typeclass property ensuring that there is no boundary (this is for
instance the case for model_with_corners_self, or products of these). Note that one could consider
as a natural assumption to only use the trivial model with corners `model_with_corners_self ℝ E`

,
but again in product manifolds the natural model with corners will not be this one but the product
one (and they are not defeq as `(λp : E × F, (p.1, p.2))`

is not defeq to the identity). So, it is
important to use the above incantation to maximize the applicability of theorems.

## Implementation notes

We want to talk about manifolds modelled on a vector space, but also on manifolds with
boundary, modelled on a half space (or even manifolds with corners). For the latter examples,
we still want to define smooth functions, tangent bundles, and so on. As smooth functions are
well defined on vector spaces or subsets of these, one could take for model space a subtype of a
vector space. With the drawback that the whole vector space itself (which is the most basic
example) is not directly a subtype of itself: the inclusion of `univ : set E`

in `set E`

would
show up in the definition, instead of `id`

.

A good abstraction covering both cases it to have a vector
space `E`

(with basic example the Euclidean space), a model space H````
(with basic example the upper
half space), and an embedding of
```

H`into`

E`(which can be the identity for`

H = E```
, or
```

subtype.val`for manifolds with corners). We say that the pair`

(E, H)```
with their embedding is a
model with corners, and we encompass all the relevant properties (in particular the fact that the
image of
```

H`in`

E`should have unique differentials) in the definition of`

model_with_corners`.

We concentrate on `C^∞`

manifolds: all the definitions work equally well for `C^n`

manifolds, but
later on it is a pain to carry all over the smoothness parameter, especially when one wants to deal
with `C^k`

functions as there would be additional conditions `k ≤ n`

everywhere. Since one deals
almost all the time with `C^∞`

(or analytic) manifolds, this seems to be a reasonable choice that
one could revisit later if needed. `C^k`

manifolds are still available, but they should be called
using `has_groupoid M (times_cont_diff_groupoid k I)`

where `I`

is the model with corners.

I have considered using the model with corners `I`

as a typeclass argument, possibly `out_param`

, to
get lighter notations later on, but it did not turn out right, as on `E × F`

there are two natural
model with corners, the trivial (identity) one, and the product one, and they are not defeq and one
needs to indicate to Lean which one we want to use.
This means that when talking on objects on manifolds one will most often need to specify the model
with corners one is using. For instance, the tangent bundle will be `tangent_bundle I M`

and the
derivative will be `mfderiv I I' f`

, instead of the more natural notations `tangent_bundle 𝕜 M`

and
`mfderiv 𝕜 f`

(the field has to be explicit anyway, as some manifolds could be considered both as
real and complex manifolds).

### Models with corners.

- to_local_equiv : local_equiv H E
- source_eq : c.to_local_equiv.source = set.univ
- unique_diff' : unique_diff_on 𝕜 (set.range c.to_local_equiv.to_fun)
- continuous_to_fun : continuous c.to_local_equiv.to_fun
- continuous_inv_fun : continuous c.to_local_equiv.inv_fun

A structure containing informations on the way a space `H`

embeds in a
model vector space `E`

over the field `𝕜`

. This is all what is needed to
define a smooth manifold with model space `H`

, and model vector space `E`

.

A vector space is a model with corners.

## Equations

- model_with_corners_self 𝕜 E = {to_local_equiv := {to_fun := id E, inv_fun := id E, source := set.univ E, target := set.univ E, map_source' := _, map_target' := _, left_inv' := _, right_inv' := _}, source_eq := _, unique_diff' := _, continuous_to_fun := _, continuous_inv_fun := _}

## Equations

- model_with_corners.has_coe_to_fun = {F := λ (e : model_with_corners 𝕜 E H), H → E, coe := λ (e : model_with_corners 𝕜 E H), e.to_local_equiv.to_fun}

The inverse to a model with corners, only registered as a local equiv.

## Equations

- I.symm = I.to_local_equiv.symm

In the trivial model with corners, the associated local equiv is the identity.

Given two model_with_corners `I`

on `(E, H)`

and `I'`

on `(E', H')`

, we define the model with
corners `I.prod I'`

on `(E × E', H × H')`

. This appears in particular for the manifold structure on
the tangent bundle to a manifold modelled on `(E, H)`

: it will be modelled on `(E × E, H × E)`

.

## Equations

- I.prod I' = {to_local_equiv := {to_fun := λ (p : H × H'), (⇑I p.fst, ⇑I' p.snd), inv_fun := λ (p : E × E'), (⇑(I.symm) p.fst, ⇑(I'.symm) p.snd), source := set.univ (H × H'), target := (set.range ⇑I).prod (set.range ⇑I'), map_source' := _, map_target' := _, left_inv' := _, right_inv' := _}, source_eq := _, unique_diff' := _, continuous_to_fun := _, continuous_inv_fun := _}

Special case of product model with corners, which is trivial on the second factor. This shows up as the model to tangent bundles.

## Equations

- I.tangent = I.prod (model_with_corners_self 𝕜 E)

Property ensuring that the model with corners `I`

defines manifolds without boundary.

The trivial model with corners has no boundary

## Equations

- _ = _

If two model with corners are boundaryless, their product also is

## Equations

- _ = _

### Smooth functions on models with corners

Given a model with corners `(E, H)`

, we define the groupoid of `C^n`

transformations of `H`

as
the maps that are `C^n`

when read in `E`

through `I`

.

Inclusion of the groupoid of `C^n`

local diffeos in the groupoid of `C^m`

local diffeos when
`m ≤ n`

The groupoid of `0`

-times continuously differentiable maps is just the groupoid of all
local homeomorphisms

An identity local homeomorphism belongs to the `C^n`

groupoid.

The composition of a local homeomorphism from `H`

to `M`

and its inverse belongs to
the `C^n`

groupoid.

### Smooth manifolds with corners

- to_has_groupoid : has_groupoid M (times_cont_diff_groupoid ⊤ I)

Typeclass defining smooth manifolds with corners with respect to a model with corners, over a
field `𝕜`

and with infinite smoothness to simplify typeclass search and statements later on.

For any model with corners, the model space is a smooth manifold

### Extended charts

In a smooth manifold with corners, the model space is the space `H`

. However, we will also
need to use extended charts taking values in the model vector space `E`

. These extended charts are
not `local_homeomorph`

as the target is not open in `E`

in general, but we can still register them
as `local_equiv`

.

The preferred extended chart on a manifold with corners around a point `x`

, from a neighborhood
of `x`

to the model vector space.

## Equations

- ext_chart_at I x = (manifold.chart_at H x).to_local_equiv.trans I.to_local_equiv

Technical lemma ensuring that the preimage under an extended chart of a neighborhood of a point in the source is a neighborhood of the preimage, within a set.

Technical lemma ensuring that the preimage under an extended chart of a neighborhood of the base point is a neighborhood of the preimage, within a set.

Technical lemma ensuring that the preimage under an extended chart of a neighborhood of a point is a neighborhood of the preimage.

Technical lemma to rewrite suitably the preimage of an intersection under an extended chart, to bring it into a convenient form to apply derivative lemmas.

In the case of the manifold structure on a vector space, the extended charts are just the identity.