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arXiv · 2107.04945

Building Three-Dimensional Differentiable Manifolds Numerically

Lee Lindblom·Oliver Rinne·Nicholas W. Taylor
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Abstract

A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as non-overlapping coordinate charts on these manifolds. It uses solutions to the two- and three-dimensional biharmonic equations in a sequence of steps that increase the differentiability of the reference metrics across the interfaces between cubic regions. This method is algorithmic and has been implemented in a computer code that automatically generates these reference metrics. Examples of three-manifolds constructed in this way are presented here, including representatives from five of the eight Thurston geometrization classes, plus the well-known Hantzsche-Wendt, the Poincare dodecahedral space, and the Seifert-Weber space.

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Lee Lindblom, Oliver Rinne, Nicholas W. Taylor. 2021-07-11. Building Three-Dimensional Differentiable Manifolds Numerically. https://doi.org/10.1016/j.jcp.2022.110957

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