1. CMB Online first
|Plane Lorentzian and Fuchsian Hedgehogs|
Parts of the Brunn-Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz-Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the Fuchsian analogues of classical geometrical inequalities (analogues which are reversed as compared to classical ones).
Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality
Categories:52A40, 52A55, 53A04, 53B30
2. CMB 2013 (vol 57 pp. 401)
|Curvature of $K$-contact Semi-Riemannian Manifolds|
In this paper we characterize $K$-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat $K$-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature $\kappa=\varepsilon$, where $\varepsilon =\pm 1$ denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a $K$-contact Lorentzian manifold.
Keywords:contact semi-Riemannian structures, $K$-contact structures, conformally flat manifolds, Einstein Lorentzian-Sasaki manifolds
Categories:53C50, 53C25, 53B30