# Article

Full entry | PDF   (0.4 MB)
Keywords:
Minkowski plane; polar expression; Landsberg angle; Frenet frame
Summary:
In this paper we study the geometry of Minkowski plane and obtain some results. We focus on the curve theory in Minkowski plane and prove that the total curvature of any simple closed curve equals to the total Landsberg angle. As the result, the sum of oriented exterior Landsberg angles of any polygon is also equal to the total Landsberg angle, and when the Minkowski plane is reversible, the sum of interior Landsberg angles of any $n$-gon is $\frac{n-2}{2}$ times of the total Landsberg angle. Our results generalizes the classical results in plane geometry. We also obtain a new characterizations of Euclidean plane among Minkowski planes.
References:
[1] Bao, D., Lackey, B.: Randers surfaces whose Laplacians have completely positive symbol. Nonlinear Anal. 38 (1999), 27–40. MR 1692941 | Zbl 0945.53045
[2] Bao, D., Shen, Z.: In the volume of unit tangent spheres in a Finsler manifold. Results Math. 26 (1994), 1–17. DOI 10.1007/BF03322283 | MR 1290676
[3] Deiche, A.: Über die Finsler-Räume mit $A_i=0$. Arch. Math. (Basel) 4 (1953), 45–51. DOI 10.1007/BF01899750 | MR 0055026
[4] Rademacher, H. B.: A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328 (2004), 373–387. DOI 10.1007/s00208-003-0485-y | MR 2036326 | Zbl 1050.53063
[5] Shen, Z.: Lectures on Finsler Geometry. World Sci., Singapore, 2001. MR 1845637 | Zbl 0974.53002

Partner of