Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE FLOW-CURVATURE OF CURVES IN A GEOMETRIC SURFACE

  • Received : 2023.02.03
  • Accepted : 2023.04.26
  • Published : 2023.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

For a fixed parametrization of a curve in an orientable two-dimensional Riemannian manifold, we introduce and investigate a new frame and curvature function. Due to the way of defining this new frame as being the time-dependent rotation in the tangent plane of the standard Frenet frame, both these new tools are called flow.

Keywords

References

  1. O. L. Calin, Entropy maximizing curves, Rev. Roumaine Math. Pures Appl. 63 (2018), no. 2, 91-106.
  2. K.-S. Chou and X.-P. Zhu, The curve shortening problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. https://doi.org/10.1201/9781420035704
  3. M. Crampin and T. Mestdag, A class of Finsler surfaces whose geodesics are circles, Publ. Math. Debrecen 84 (2014), no. 1-2, 3-16. https://doi.org/10.5486/PMD.2014.5845
  4. M. Crasmareanu, A new approach to gradient Ricci solitons and generalizations, Filomat 32 (2018), no. 9, 3337-3346. https://doi.org/10.2298/FIL1809337C
  5. M. Crasmareanu, The flow-curvature of spacelike parametrized curves in the Lorentz plane, Proc. Int. Geom. Cent. 15 (2022), no. 2, 100-108. https://doi.org/10.15673/tmgc.v15i2.2281
  6. M. Crasmareanu, The flow-curvature of plane parametrized curves, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat. 72 (2023), in press.
  7. M. Crasmareanu, The flow-geodesic curvature and the flow-evolute of hyperbolic plane curves, Int. Electron. J. Geom. 16 (2023), no. 1, 225-231. https://doi.org/10.36890/iejg.1229215
  8. M. Crasmareanu, The flow-geodesic curvature and the flow-evolute of spherical curves, submitted.
  9. M. Crasmareanu and C. Frigioiu, Unitary vector fields are Fermi-Walker transported along Rytov-Legendre curves, Int. J. Geom. Methods Mod. Phys. 12 (2015), no. 10, 1550111, 9 pp. https://doi.org/10.1142/S021988781550111X
  10. Y. Fang, The Minding formula and its applications, Arch. Math. (Basel) 72 (1999), no. 6, 473-480. https://doi.org/10.1007/s000130050358
  11. B. C. Mazur, Perturbations, deformations, and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 307-336. https://doi.org/10.1090/S0273-0979-04-01024-9
  12. P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge Univ. Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511609565
  13. B. O'Neill, Elementary Differential Geometry, revised second edition, Elsevier/Academic Press, Amsterdam, 2006.