feat(Geometry/Manifold): orientable manifolds

[michaellee94]

---

• depends on: feat(Geometry/Manifold): the interior of a manifold is open #33189
  

[github-actions]

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[github-actions]

PR summary 7314c8ed3a

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Geometry.Manifold.Instances.Real	2271	2284	+13 (+0.57%)
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary	2201	2208	+7 (+0.32%)

Import changes for all files

Files	Import difference
Mathlib.Geometry.Manifold.WhitneyEmbedding	2
Mathlib.Geometry.Manifold.Riemannian.Basic Mathlib.Geometry.Manifold.Riemannian.PathELength	3
Mathlib.Geometry.Manifold.Bordism Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary	7
3 files Mathlib.Geometry.Manifold.Instances.Sphere Mathlib.Geometry.Manifold.PoincareConjecture Mathlib.Topology.Compactification.OnePoint.Sphere	8
Mathlib.Geometry.Manifold.Instances.Icc	10
Mathlib.Geometry.Manifold.Instances.Real	13
Mathlib.Geometry.Manifold.Orientation (new file)	2282

---

Declarations diff

+ IccInteriorOpens
+ ManifoldOrientation
+ ManifoldOrientation.compatible_and_locallyConstant
+ ManifoldOrientation.ext_sign
+ ManifoldOrientation.tangentOrientation
+ ManifoldOrientation.tangentOrientation_eq_positive_or_neg
+ Orientable
+ OrientationPreservingOn
+ OrientedManifold
+ _root_.DifferentiableAt.mem_interior_convex_of_surjective_fderiv
+ baseOrientation
+ coe_IccInteriorOpens_eq_interior
+ contDiffOn_extCoordChange
+ contDiffOn_extCoordChange_symm
+ extCoordChange
+ extCoordChange_symm
+ instHasGroupoidIccInteriorOrientation
+ instOrientableIcc
+ instOrientedManifoldIcc
+ instance (M : Type*) [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]
+ instance : ClosedUnderRestriction (orientationPreservingGroupoid I)
+ instance {n : WithTop ℕ∞} [IsManifold I n M] (hn : n ≠ 0) :
+ interiorOpens
+ isBoundaryPoint_iff_of_mem_atlas
+ isClosed_boundary
+ isInteriorPoint_iff_of_mem_atlas
+ isInvertible_fderivWithin_extCoordChange
+ isOpen_interior
+ jacobianDetWithin
+ mem_frontier_iff_notMem_interior
+ mem_interior_iff_notMem_frontier
+ mem_orientationPreservingGroupoid_iff
+ modelSet
+ modelSet_inter
+ modelSet_uniqueDiffOn
+ ofSet_mem_orientationPreservingGroupoid
+ orientationFromSign
+ orientationPreservingGroupoid
+ orientationPreservingGroupoid_le_contDiffGroupoid
+ orientationPreservingOn_ofSet
+ orientationPreservingPregroupoid
+ orientedTangentOrientation
+ point_has_two_manifoldOrientations
+ uniqueDiffOn_extCoordChange_source
+ uniqueDiffOn_extCoordChange_target

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(Geometry/Manifold): the interior of a manifold is open #33189
  By Dependent Issues (🤖). Happy coding!