Opened 6 months ago

Last modified 3 months ago

#31707 new enhancement

Manifold of piecewise linear functions with k marked breakpoints

Reported by: mkoeppe Owned by:
Priority: major Milestone: sage-9.5
Component: manifolds Keywords:
Cc: yzh, egourgoulhon, gh-mjungmath, tscrim Merged in:
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Report Upstream: N/A Work issues:
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Description

We define the Riemannian manifold PL_k of continuous piecewise linear functions from the interval [0,1] to the reals with k+1 marked breakpoints.

Default chart: breakpoints 0 = a_0 < a_1 < a_2 < ... < a_k = 1; slopes s_1, ..., s_k.

The breakpoints are marked: For example for k = 2, we distinguish the constant function with a_0 = 0, a_1 = 1/2, a_2 = 1 and s_1 = s_2 = 0 from the constant function with a_1 = 1/3.

This manifold has an immersion (but not embedding) into the Hilbert space of L2 functions. The inner product there (see #30218) pulls back to define the metric on PL_k.

Elements of PL_k indicate their embedding.

Change History (2)

comment:1 Changed 6 months ago by mkoeppe

  • Cc tscrim added

Short of defining infinite-dimensional Banach and Hilbert manifolds, for this ticket we would define

  • a version of ContinuousMap that maps into an arbitrary (topological) VectorSpace (with distinguished basis)
  • a version of DiffMap that maps into an arbitrary (topological) VectorSpace or InnerProductSpace (#30218) (with distinguished basis)
  • versions of TopologicalSubmanifold, DifferentiableSubmanifold, PseudoRiemannianSubmanifold that can work with these types of maps.

comment:2 Changed 3 months ago by mkoeppe

  • Milestone changed from sage-9.4 to sage-9.5
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