Opened 22 months ago
Last modified 4 weeks ago
#31728 new enhancement
manifolds.Sphere: Make relation to simplicial spheres more concrete
Reported by: | mkoeppe | Owned by: | |
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Priority: | major | Milestone: | sage-9.9 |
Component: | manifolds | Keywords: | |
Cc: | gh-mjungmath, yzh, gh-kliem, egourgoulhon | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description
The example manifolds.Sphere
defines a method minimal_triangulation
, which returns a simplicial sphere (the boundary of the (n+1)-simplex) as an abstract simplicial complex.
We propose to make the relationship between the two objects more concrete by introducing intermediate objects and maps as follows.
Define a geometric realization of the simplicial complex as the polyhedral complex that is the boundary of a geometric (n+1)-simplex (or of any full-dimensional polyhedron).
Define the face_manifold_poset
(#31660) of the polyhedron. The poset elements are images of embedded submanifolds (and subsets) of the Euclidean space.
Let c be a point in the interior of the polyhedron.
Define the differentiable map sending En+1 \ c to En+1 \ 0, defined in Cartesian coordinates as x ⟼ (x-c)/|x-c|. It pulls back to differentiable maps on the embedded submanifolds, defining differentiable embeddings into the sphere; and to a continuous map on their union, defining a continuous embedding into the sphere.
Change History (5)
comment:1 Changed 19 months ago by
Milestone: | sage-9.4 → sage-9.5 |
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comment:2 Changed 14 months ago by
Milestone: | sage-9.5 → sage-9.6 |
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comment:3 Changed 11 months ago by
Milestone: | sage-9.6 → sage-9.7 |
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comment:4 Changed 5 months ago by
Milestone: | sage-9.7 → sage-9.8 |
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comment:5 Changed 4 weeks ago by
Milestone: | sage-9.8 → sage-9.9 |
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