Legendre curves on 3-dimensional $C_{12}$-Manifolds

Abstract

Legendre curves play a very important and special role in geometry and topology of almost contact manifolds.There are certain results known for Legendre curves in 3-dimensional normal almost contact manifolds. The aim of this paper is to study Legendre curves of three-dimensional $C_{12}$-manifolds which are non-normal almost contact manifolds and classifying all biharmonic Legendre curves in these manifolds.

Lorentz ${C_{12}}$-manifolds

“…It is known that the almost contact structure (φ, ξ, η) is said to be normal if and only if N (1) (X, Y ) = N φ (X, Y ) + 2dη(X, Y )ξ = 0 for any X, Y on M , where N φ denotes the Nijenhuis torsion of φ, given by…”

“…Recently, C 12 -manifolds have become a well-known and intensively studied subject of research in differential geometry. The recent works [1,2,3,4,5] provide a detailed overview of the results obtained in this framework.…”

Ricci Soliton in Deformed C12-Manifolds

“…Recent scholarly contributions have emerged, delving into a distinctive category of almost contact metric manifolds termed C 12 -manifolds. These manifolds, unlike their normal or contact counterparts, have garnered attention in works such as those presented in [1,3,4,5]. Hamilton, in 1982 [8, 9], introduced the concept of Ricci flow as a method to establish a canonical metric on smooth manifolds.…”

Characterization of 3-dimensional almost contact metric structures

“…For the class C 12 which is integrable and never normal, recently some works have been published on this subject. For example, [1,2,3,9]. But, for the class C 9 which is neither normal nor integrable, unfortunately, there is no study on it yet.…”