Differential Equations of Spacelike Parametrized Curves in the Lorentz Plane

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

1. Abe, N., Nakanishi, Y., Yamaguchi, S., Circles and spheres in pseudo-Riemannian geometry, Aequationes Math., 39(2-3)(1990), 134–145.
2. 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.
3. 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.
4. Crasmareanu, M., The flow-curvature of spacelike parametrized curves in the Lorentz plane, Proc. Int. Geom. Cent., 15(2)(2022), 101–109.
5. Crasmareanu, M., The adjoint map of Euclidean plane curves and curvature problems, Tamkang J. Math., 55(2024), (in press).
6. Saloom, A., Tari, F., Curves in the Minkowski plane and their contact with pseudo-circles, Geom. Dedicata, 159(2012), 109–124.
7. Olver Peter J., Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995.
8. Woolgar, E., Xie, R., Self-similar curve shortening flow in hyperbolic 2-space, Proc. Am. Math. Soc., 150(3)(2022), 1301–1319.