On the four vertex theorem in planes with radial density \(e^{\varphi(r)}\). (English) Zbl 1152.53042

Doan The Hieu; Tran Le Nam

Colloq. Math. 113, No. 1, 169-174 (2008).

The authors study the four vertex theorem in a special 2-dimensional Riemannian manifold, namely the plane with radial density \(\exp\varphi(r)\), where \(r\) is the distance from the origin and \(\varphi\) is a given function of class \(C^2\). In this manifold a curve \(c\in C^3\) with Euclidean curvature \(k\) has the “\(\varphi\)-curvature” \(k_\varphi= k- d\varphi/dn\) (\(n\): normal unit vector of \(c\)).
It is shown that in a plane with radial density \(\exp\varphi(r)\) the following main results hold: Every simple closed curve that is invariant under a rotation about the origin by an angle \(\theta <\pi\) has at least four vertices (Corollary 2); The four vertex theorem holds for the class of all simple closed curves if and only if \(\varphi\) is a constant (Theorem 3); For every positive integer \(n\geq 1\), there exists a radial density \(\exp\varphi(r)\) in the plane such that a given circle containing the origin in its interior, but not as the centre, has exactly \(2n\) vertices (Corollary 6).

Reviewer: Richard Koch (München)

MSC:

53C40	Global submanifolds
53A04	Curves in Euclidean and related spaces

Keywords:

Euclidean plane; radial density; simple closed curve; four vertex theorem; circle; curvature with density; plane with radial density