Irregularities in the perimeter of a water droplet adhering to a vertical pane of glass cause perturbations in the curvature of the droplet surface. When laser light passes through such a droplet, the perturbations produce a far field refraction caustic, which is a section of the caustic known as the parabolic umbilic in the catastrophe theory classification. As the water evaporates and the dro...

In this paper, we establish some inequalities between the normalized δ-Casorati curvatures and scalar curvature (i.e., extrinsic intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like manifolds, endowed with a semi-symmetric metric connection. Moreover, study satisfying equality cases these inequalities. We also present an appropriate example.

After early work of Hénon it has become folk knowledge that symmetric periodic orbits are of particular importance. We reinforce this belief by additional studies and we further find that invariant closed symplectic submanifolds caused by discrete symmetries prove to be an important complement to the long known role of the orbits. The latter have particular importance in semi-classics. Based on...

‎we prove that there do not exist totally contact umbilical‎ ‎proper slant lightlike submanifolds of indefinite sasakian manifolds other than totally contact geodesic‎ ‎proper slant lightlike submanifolds‎. ‎we also prove that there do‎ ‎not exist totally contact umbilical proper slant lightlike‎ ‎submanifolds of indefinite sasakian space forms‎.

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

* Correspondence: meraj79@gmail. com Department of Mathematics, University of Tabuk, Tabuk, Kingdom of Saudi Arabia Full list of author information is available at the end of the article Abstract In this article, we have obtained necessary and sufficient conditions in terms of canonical structure F on a semi-invariant submanifold of an almost contact manifold under which the submanifold reduced...