Complex Line Bundles Over Simplicial Complexes and Their Applications

Felix Knöppel () and Ulrich Pinkall ()
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Felix Knöppel: Technische Universität Berlin, Inst. für Mathematik
Ulrich Pinkall: Technische Universität Berlin, Inst. für Mathematik

A chapter in Advances in Discrete Differential Geometry, 2016, pp 197-239 from Springer

Abstract: Abstract Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise-constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension.

Keywords: Simplicial Complex; Hermitian Line Bundle; Vector Bundle; Dirichlet Energy; Piecewise Constant Curvature (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-50447-5_6

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DOI: 10.1007/978-3-662-50447-5_6

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