feat(Geometry/Manifold): continuously differentiable actions

[archiebrowne]

define the class ContDiffSMul 𝕜 M X n which asserts that the map (c, x) ↦ c • x is n times continuously differentiable on M × X.

Many of the results are the C^n analogues of those in the module Mathlib.Topology.Algebra.MulAction.

[github-actions]

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[github-actions]

PR summary 452dba9608

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.ContDiffMulAction (new file)	1795

---

Declarations diff

+ ContDiff.contdiff_smul
+ ContDiffAt.contdiff_smul
+ ContDiffOn.contdiff_smul
+ ContDiffSMul
+ ContDiffSMul.comp
+ ContDiffSMul.of_le
+ ContDiffSMul.of_retraction
+ ContDiffSMul.op
+ ContDiffSMul.toContinuousSMul
+ ContDiffVAdd
+ ContDiffWithinAt.contdiff_smul'
+ ContinuousLinearEquiv.contDiffSMul
+ ContinuousLinearMap.contDiffSMul
+ Filter.Tendsto.contdiff_smul
+ Filter.Tendsto.contdiff_smul_const
+ IsScalarTower.contdiffSMul
+ MulAction.contDiffSMul_compHom
+ MulOpposite.contDiffSMul
+ MulOpposite.contDiff_op
+ MulOpposite.contDiff_unop
+ NormedAlgebra.contDiffSMul_self
+ NormedSpace.contDiffSMul
+ Prod.contDiffSMul
+ instance {ι : Type*} [Fintype ι] {γ : ι → Type*} [∀ i, SMul M (γ i)]

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[peabrainiac]

I just found this PR while preparing a PR on continuously differentiable actions of manifolds (i.e. ContMDiffSMul instead of ContDiffSMul) and looking for prior work. In principle I think having a non-manifold version too makes sense; the acting space being a vector space is a strong restriction, but there are at least some examples of such actions that are not just the action of a field on a vector space over it, e.g. the action of the monoid of all continuous linear transformations of a space on that space. The API for non-manifold actions should probably go into Mathlib/Analysis though, not in the Mathlib/Geometry/Manifold folder.

I've left a few other remarks below.

[peabrainiac]

The correct name for this would be ContDiff.smul, but that name is already taken by the corresponding lemma for scalar multiplication on vector spaces - if possible without messing with the import structure too much, it would probably make sense to remove those lemmas and provide a ContDiffSMul instance for scalar multiplication on vector spaces instead, and then use the correct name here? The same goes for ContDiffAt.contdiff_smul etc.

[archiebrowne]

Thanks! I'll have a think about how to do this cleanly

[archiebrowne]

I just found this PR while preparing a PR on continuously differentiable actions of manifolds (i.e. ContMDiffSMul instead of ContDiffSMul) and looking for prior work. In principle I think having a non-manifold version too makes sense; the acting space being a vector space is a strong restriction, but there are at least some examples of such actions that are not just the action of a field on a vector space over it, e.g. the action of the monoid of all continuous linear transformations of a space on that space. The API for non-manifold actions should probably go into Mathlib/Analysis though, not in the Mathlib/Geometry/Manifold folder.

I've left a few other remarks below.

Thanks for the review! I have committed most of the suggestions.