The derivative of functions between smooth manifolds

Let M and M' be two smooth manifolds with corners over a field 𝕜 (with respective models with corners I on (E, H) and I' on (E', H')), and let f : M → M'. We define the derivative of the function at a point, within a set or along the whole space, mimicking the API for (Fréchet) derivatives. It is denoted by mfderiv I I' f x, where "m" stands for "manifold" and "f" for "Fréchet" (as in the usual derivative fderiv 𝕜 f x).

Main definitions

• unique_mdiff_on I s : predicate saying that, at each point of the set s, a function can have at most one derivative. This technical condition is important when we define mfderiv_within below, as otherwise there is an arbitrary choice in the derivative, and many properties will fail (for instance the chain rule). This is analogous to unique_diff_on 𝕜 s in a vector space.

Let f be a map between smooth manifolds. The following definitions follow the fderiv API.

• mfderiv I I' f x : the derivative of f at x, as a continuous linear map from the tangent space at x to the tangent space at f x. If the map is not differentiable, this is 0.
• mfderiv_within I I' f s x : the derivative of f at x within s, as a continuous linear map from the tangent space at x to the tangent space at f x. If the map is not differentiable within s, this is 0.
• mdifferentiable_at I I' f x : Prop expressing whether f is differentiable at x.
• mdifferentiable_within_at 𝕜 f s x : Prop expressing whether f is differentiable within s at x.
• has_mfderiv_at I I' f s x f' : Prop expressing whether f has f' as a derivative at x.
• has_mfderiv_within_at I I' f s x f' : Prop expressing whether f has f' as a derivative within s at x.
• mdifferentiable_on I I' f s : Prop expressing that f is differentiable on the set s.
• mdifferentiable I I' f : Prop expressing that f is differentiable everywhere.
• tangent_map I I' f : the derivative of f, as a map from the tangent bundle of M to the tangent bundle of M'.

We also establish results on the differential of the identity, constant functions, charts, extended charts. For functions between vector spaces, we show that the usual notions and the manifold notions coincide.

Implementation notes

The tangent bundle is constructed using the machinery of topological fiber bundles, for which one can define bundled morphisms and construct canonically maps from the total space of one bundle to the total space of another one. One could use this mechanism to construct directly the derivative of a smooth map. However, we want to define the derivative of any map (and let it be zero if the map is not differentiable) to avoid proof arguments everywhere. This means we have to go back to the details of the definition of the total space of a fiber bundle constructed from core, to cook up a suitable definition of the derivative. It is the following: at each point, we have a preferred chart (used to identify the fiber above the point with the model vector space in fiber bundles). Then one should read the function using these preferred charts at x and f x, and take the derivative of f in these charts.

Due to the fact that we are working in a model with corners, with an additional embedding I of the model space H in the model vector space E, the charts taking values in E are not the original charts of the manifold, but those ones composed with I, called extended charts. We define written_in_ext_chart I I' x f for the function f written in the preferred extended charts. Then the manifold derivative of f, at x, is just the usual derivative of written_in_ext_chart I I' x f, at the point (ext_chart_at I x) x.

There is a subtelty with respect to continuity: if the function is not continuous, then the image of a small open set around x will not be contained in the source of the preferred chart around f x, which means that when reading f in the chart one is losing some information. To avoid this, we include continuity in the definition of differentiablity (which is reasonable since with any definition, differentiability implies continuity).

Warning: the derivative (even within a subset) is a linear map on the whole tangent space. Suppose that one is given a smooth submanifold N, and a function which is smooth on N (i.e., its restriction to the subtype N is smooth). Then, in the whole manifold M, the property mdifferentiable_on I I' f N holds. However, mfderiv_within I I' f N is not uniquely defined (what values would one choose for vectors that are transverse to N?), which can create issues down the road. The problem here is that knowing the value of f along N does not determine the differential of f in all directions. This is in contrast to the case where N would be an open subset, or a submanifold with boundary of maximal dimension, where this issue does not appear. The predicate unique_mdiff_on I N indicates that the derivative along N is unique if it exists, and is an assumption in most statements requiring a form of uniqueness.

On a vector space, the manifold derivative and the usual derivative are equal. This means in particular that they live on the same space, i.e., the tangent space is defeq to the original vector space. To get this property is a motivation for our definition of the tangent space as a single copy of the vector space, instead of more usual definitions such as the space of derivations, or the space of equivalence classes of smooth curves in the manifold.

Tags

Derivative, manifold

Derivative of maps between manifolds

The derivative of a smooth map f between smooth manifold M and M' at x is a bounded linear map from the tangent space to M at x, to the tangent space to M' at f x. Since we defined the tangent space using one specific chart, the formula for the derivative is written in terms of this specific chart.

We use the names mdifferentiable and mfderiv, where the prefix letter m means "manifold".

Unique differentiability sets in manifolds

General lemmas on derivatives of functions between manifolds

We mimick the API for functions between vector spaces

Deriving continuity from differentiability on manifolds