The Pólya f-curvature of plane curves
DOI:
https://doi.org/10.24193/subbmath.2024.1.14Keywords:
Plane curve, Pólya vector field, f-curvature, reverse potentialAbstract
We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it Pólya, being the difference between the usual curvature and the inner product of the normal vector field with the Pólya vector field of a given planar function f. We computed it for several examples, since the general problem of vanishing or constant values of this new curvature involves the general expression of f.
Mathematics Subject Classification (2010): 53A04, 53A45, 53A55.
Received 03 May 2023; Accepted 17 January 2024
References
Alexanderson, G.L., The Random Walks of George Pólya, MAA Spectrum, Washington, DC, Mathematical Association of America (MAA), 2000.
Barbosa, E., Santana, F., Upadhyay, A., λ-Hypersurfaces on shrinking gradient Ricci solitons, J. Math. Anal. Appl., 492(2020), no. 2, Article ID 124470, 18 p.
Braden, B., Pólya’s geometric picture of complex contour integrals, Math. Mag., 60(1987), 321-327.
Chou, K.S., Zhu, X.P., The Curve Shortening Problem, Boca Raton, FL, Chapman & Hall/CRC, 2001.
Crășmăreanu, M., New tools in Finsler geometry: Stretch and Ricci solitons, Math. Rep., Bucharest, 16(66)(2014), no. 1, 83-93.
Crășmăreanu, M., Para-mixed linear spaces, Acta Univ. Sapientiae Math., 9(2017), no. 2, 275-282.
Crășmăreanu, M., Frigioiu, C., Unitary vector fields are Fermi-Walker transported along Rytov-Legendre curves, Int. J. Geom. Methods Mod. Phys., 12(10)(2015), Article ID 1550111, 9 pages.
Gromov, M., Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13(2003), no. 1, 178-215; erratum ibid., 18(2008), no. 5, 1786-1786.
Ionescu, L.M., Pripoae, C.L., Pripoae, G.T., Classification of holomorphic functions as Pólya vector fields via differential geometry, Mathematics, 9(16)(2021), article no. 1890.
Klingenberg, W., A Course in Differential Geometry, Springer, 1978.
Mazur, B., Perturbations, deformations, and variations in geometry, physics, and number theory, Bull. Am. Math. Soc., New Ser., 41(3)(2004), 307-336.
Miron, R., Une généralisation de la notion de courbure de parallélisme, Gaz. Mat. Fiz., București, Ser. A, 10(63)(1958), 705-708.
Miron, R., The Geometry of Myller Configurations. Applications to Theory of Surfaces and Nonholonomic Manifolds, Bucharest, Editura Academiei Române, 2010.
Zhu, C.P., Lectures on Mean Curvature Flows, AMS/IP Studies in Advanced Mathematics, 32, Providence, RI, American Mathematical Society, 2002.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
