Differential Equations of Spacelike Parametrized Curves in the Lorentz Plane

Year 2024, , 1 - 5, 30.06.2024

Mircea Crasmareanu

https://doi.org/10.47000/tjmcs.1403706

Abstract

We introduce four ordinary differential equations for a fixed natural parametrization of a spacelike curve $C$ in the Lorentz plane. The relationships between these differential equations is studied through the curvature of $C$.

Keywords

Lorentz plane, spacelike curve, curvature, differential equations.

References

• Abe, N., Nakanishi, Y., Yamaguchi, S., Circles and spheres in pseudo-Riemannian geometry, Aequationes Math., 39(2-3)(1990), 134–145.
• Castro, I., Castro-Infantes, I., Castro-Infantes, J., Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers, Open Math. 16(2018), 747–766.
• Castro, I., Castro-Infantes, I., Castro-Infantes, J., Curves in the Lorentz-Minkowski plane with curvature depending on their position, Open Math., 18(2020), 749–770.
• Crasmareanu, M., The flow-curvature of spacelike parametrized curves in the Lorentz plane, Proc. Int. Geom. Cent., 15(2)(2022), 101–109.
• Crasmareanu, M., The adjoint map of Euclidean plane curves and curvature problems, Tamkang J. Math., 55(2024), (in press).
• Saloom, A., Tari, F., Curves in the Minkowski plane and their contact with pseudo-circles, Geom. Dedicata, 159(2012), 109–124.
• Olver Peter J., Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995.
• Woolgar, E., Xie, R., Self-similar curve shortening flow in hyperbolic 2-space, Proc. Am. Math. Soc., 150(3)(2022), 1301–1319.