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On the four vertex theorem in planes with radial density e φ ( r )

Doan The Hieu, Tran Le Nam (2008)

Colloquium Mathematicae

It is shown that in a plane with a radial density the four vertex theorem holds for the class of all simple closed curves if and only if the density is constant. On the other hand, for the class of simple closed curves that are invariant under a rotation about the origin, the four vertex theorem holds for every radial density.

On the geometry of frame bundles

Kamil Niedziałomski (2012)

Archivum Mathematicum

Let ( M , g ) be a Riemannian manifold, L ( M ) its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle T M . We compute the Levi–Civita connection and curvatures of these metrics.

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