Research and Innovation Articles
Published 2020-01-24
Keywords
- Gaussian curvature,
- continuous binomial coefficients
How to Cite
Cano G., L. A., & Carrillo, S. A. (2020). Can we detect Gaussian curvature by counting paths and measuring their length?. Revista Integración, Temas De matemáticas, 38(1), 33–42. https://doi.org/10.18273/revint.v38n1-2020003
Abstract
The aim of this paper is to associate a measure for certain sets of
paths in the Euclidean plane R2 with fixed starting and ending points. Then,
working on parameterized surfaces with a specific Riemannian metric, we
define and calculate the integral of the length over the set of paths obtained
as the image of the initial paths in R2 under the given parameterization.
Moreover, we prove that this integral is given by the average of the lengths
of the external paths times the measure of the set of paths if, and only if, the
surface has Gaussian curvature equal to zero.
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References
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Abramowitz M. and Stegun I., Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, Dover, New York, ninth Dover printing, tenth GPO
printing edition, 1964.
Cano L. and Díaz R., “Indirect Influences on Directed Manifolds”, Adv. Stud. Contemp.
Math. 28 (2018), No. 1, 93–114.
Cano L. and Díaz R., “Continuous Analogues for the Binomial Coefficients and the Catalan
Numbers”, arXiv:1602.09132.
Chaichian M. and Demichev A., Path Integrals in Physics: Volume I Stochastic Processes
and Quantum Mechanics, Series in Mathematical and Computational Physics, Taylor &
Francis, 2001.
Feynman R. and Hibbs A., Quantum mechanics and integrals, McGraw–Hill Companies,
Inc., New York, 1985.
Salwinski D., “The Continuous Binomial Coefficient: An Elementary Approach”, Amer.
Math. Monthly 123 (2018), No. 3, 231–244.
Wakhare T. and Vignat C., “A continuous analogue of lattice path enumeration: part II”,
Online J. Anal. Comb 14 (2019), #04, 22 p.