Manifold of piecewise linear functions with k marked breakpoints

[mkoeppe]

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.

CC: @yuan-zhou @egourgoulhon @mjungmath @tscrim

Component: manifolds

Issue created by migration from https://trac.sagemath.org/ticket/31707

[mkoeppe]

comment:1

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.