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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.
The example
manifolds.Sphere
defines a methodminimal_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.
CC: @mjungmath @yuan-zhou @kliem @egourgoulhon
Component: manifolds
Issue created by migration from https://trac.sagemath.org/ticket/31728
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