Proper Helix of Order 6 And LC Helix in Pseudo-Euclidean Space E84

Abstract

In this paper, we used the result that complex hyperbolic spaces CH²(−4c/3 ) with holomorphic sectional curvature −4c/3 are isometrically embedded in E⁸₄ . By considering a circle in CH²(−4c/3 ), we prove that the image of the circle by isometric embedding is a proper helix of order 6 in E⁸₄ . Moreover, we define a generalized LC helix on a submanifold of E⁸₄ . Also, we show that the image of a circle by isometric embedding from complex hyperbolic plane CH²(−4c/3 ) to pseudo-Euclidean space E⁸₄ is a generalized LC helix on some submanifold of E⁸₄ .

Key words and phrases. Curve of proper order, complex hyperbolic space, generalized LC helix, isometric embedding, parallel second fundamental form.

2010 Mathematics Subject Classification. 53C50, 53C40