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CURVATURE TESTING IN 3-DIMENSIONAL METRIC POLYHEDRAL COMPLEXES
 

Summary: CURVATURE TESTING IN 3-DIMENSIONAL
METRIC POLYHEDRAL COMPLEXES
MURRAY ELDER AND JON MCCAMMOND1
Abstract. In a previous article, the authors described an algorithm
to determine whether a finite metric polyhedral complex satisfied vari-
ous local curvature conditions such as being locally CAT(0). The proof
made use of Tarski's theorem about the decidability of first order sen-
tences over the reals in an essential way and thus was not immediately
applicable to a specific finite complex. In this article we describe an algo-
rithm restricted to 3-dimensional complexes which uses only elementary
3-dimensional geometry. After describing the procedure we include sev-
eral examples involving Euclidean tetrahedra which were run using an
implementation of the algorithm in GAP.
In this article we describe an algorithm to determine whether or not a
finite 3-dimensional M-complex is locally CAT(). The procedure is based
purely on elementary 3-dimensional geometry, and has considerable compu-
tational advantages over the algorithm for M-complexes of arbitrary dimen-
sion which was described by the authors in [6]. After describing the proce-
dure we include several examples involving Euclidean tetrahedra which were
run using an implementation of the algorithm in GAP. The implementation

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics