Frenet frames and invariants of timelike ruled surfaces

On closed timelike and spacelike ruled surfaces

“…The characterization of a ruled surface is given as follows: M is timelike, if is spacelike M is spacelike, if is timelike . …”

Spacelike Regle Yüzeylerin Frenet Çatıları ve Frenet İnvaryantları

“…Because of this position of the ruled surfaces, many geometers have studied on them in the Euclidean space and they have investigated many properties of the ruled surfaces [6,7,12,14]. Furthermore, the differential geometry of the ruled surfaces in Minkowski space has been studied by several authors [2][3][4]8,9,11,15].…”

“…Moreover, Önder and Uğurlu have introduced the Frenet frames, Frenet invariants and the instantaneous rotation vectors of timelike ruled surfaces in the Minkowski 3-space [11].…”

Evolutes of Hyperbolic Dual Spherical Curve in Dual Lorentzian 3-Space

“…There are some articles concerning singularities of surfaces and classical geometric invariants of space curves for several kinds of geometry [13][14][15][16][17][18][19][20][21][22][23]. In these articles the corresponding functions depend on each geometry.…”

“…One of the main techniques for applying the singularity theory to Euclidean differential geometry is to consider the distance squared function and the height function on a submanifold of E 3 [13,14]. There are some articles concerning singularities of surfaces and classical geometric invariants of space curves for several kinds of geometry [13][14][15][16][17][18][19][20][21][22][23].…”