Kurventheorie im konform isotropen RaumC 3 (1)

LetM be a 3-dimensional manifold with metric tensorg. (M,g) is called a conformally isotropce space C ₃ ⁽¹⁾ if there exists a chart (M, ) of M, for which (i)(M)=³, (ii) the components ofg with respect to areg ₁₁=g ₂₂=q,g ₁₂=g ₁₃=g ₂₃=g ₃₃=0q(x ¹,x²,x³) > 0, . In this note, first we consider some metric properties ofC ₃ ⁽¹⁾ . Further it is shown that there exists a unique linear connection inC ₃ ⁽¹⁾ , the so-called standard connection. Finally we develop the fundamentals of the theory of curves inC ₃ ⁽¹⁾ up to Frenet's formula, and give a geometric interpretation of the conformally isotropic curvature resp. torsion of a curvec inC ₃ ⁽¹⁾ . It is shown, that , where resp. are the isotropic curvature resp. isotropic torsion of the curvec in the threedimensional isotropic spaceI ₃ ⁽¹⁾ when using a special coordinate system ofC ₃ ⁽¹⁾ as the standard coordinate system ofI ₃ ⁽¹⁾ .

---

---

Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet