The equation of Euler's line yields a Tzitzeica surface

Abstract

at Fullerton. His main fields of research include affine differential geometry, asymptotic analysis, twistor theory, geometric algebra and their applications in physics. Alexandru Bobe received his Ph.D. in 2007. Presently, he is an assistant professor at the Ovidius University in Constanţa, Romania. His main fields of research include affine differential geometry, Riemannian geometry, computational algebra and geometry.

Tzitzeica-Type centro-affine invariants in Minkowski spaces

“…It is easy to see that this surface is a Tzitzeica surface in the Minkowski 3-dimensional space R 3 1 , but the equation of this Minkowski sphere in the Euclidian 3-dimensional space represents an hyperboloid which is different to the Euclidian sphere. The Tzitzeica surface induced by the equation of the Euler's line of a triangle as in [2] is an example of Tzitzeica surface in a Minkowski space according to our theory, too. Then the equations of Tzitzeica surfaces in Minkowski spaces are equations of Tzitzeica surfaces in Euclidian spaces and viceversa.…”

Recovering the cosmological constant from affine geometry

“…The main results and some further conjectures can be found in [23]. Some other results on Tzitzeica surfaces in Euclidean spaces are reported in [24,25]. A general theory about curves and surfaces in Minkowski spaces is presented in [26], while the extension of the concept of Tzitzeica surfaces to Minkowski 3dimensional spaces is presented in [27].…”

Wick–Tzitzeica solitons and their Monge–Ampér equation

“…) is respectively the real and the complex part of the holomorphic function F (z) = z2 , hence there are (Euclidean) harmonic maps, i.e. belongs to the kernel of the Euclidean Laplacian ∆ := ∂ 2 ∂x 2 + ∂ 2 ∂y 2 = 4∂∂.…”