Smooth manifolds (possibly with boundary or corners) #

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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 changes 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 : ℕ∞). With this groupoid at hand and the general machinery of charted spaces, 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.
• 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 charted space 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 (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)

• model_with_corners_self ℝ (euclidean_space (fin n)) for the model space used to define n-dimensional real manifolds without boundary (with notation 𝓡 n in the locale manifold)
• model_with_corners ℝ (euclidean_space (fin n)) (euclidean_half_space n) for the model space used to define n-dimensional real manifolds with boundary (with notation 𝓡∂ n in the locale manifold)
• model_with_corners ℝ (euclidean_space (fin 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] [charted_space (euclidean_space (fin n)) M] [smooth_manifold_with_corners (𝓡 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 (fin n)) × (euclidean_space (fin n)) and not on euclidean_space (fin (2 * n))! In the same way, it would not apply to product manifolds, modelled on (euclidean_space (fin n)) × (euclidean_space (fin m)). The right invocation does not focus on one specific construction, but on all constructions sharing the right properties, like

variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] {I : model_with_corners ℝ E E} [I.boundaryless] {M : Type*} [topological_space M] [charted_space 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 (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. #

Smooth functions on models with corners #

Smooth manifolds with corners #

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.