Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^n$, $n\ge 2$, in Euclidean space satisfying a relation ${\sum }_{r=1}^n{(-1)}^{r+1}r{f}^{r-1}\left(\genfrac{}{}{0pt}{}{n}{r}\right)H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^{\infty }(M^n;ℝ)$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms of harmonic...

A Weitzenböck formula for SL(q)-foliations is derived. Its linear part is a relative trace of the relative curvature operator acting on vector valued forms.

A Weitzenböck formula for the Laplace-Beltrami operator acting on differential forms on Lie algebroids is derived.

We study the geometric structure of a Gauduchon manifold of constant curvature. We give a necessary and sufficient condition for a Gauduchon manifold to be a Gauduchon manifold of constant curvature, and we classify the Gauduchon manifolds of constant curvature. Next, we investigate Weyl submanifolds of such manifolds.

We define Weyl submersions, for which we derive equations analogous to the Gauss and Codazzi equations for an isometric immersion. We obtain a necessary and sufficient condition for the total space of a Weyl submersion to admit an Einstein-Weyl structure. Moreover, we investigate the Einstein-Weyl structure of canonical variations of the total space with Einstein-Weyl structure.

A submanifold $M^m$ of a generalized Sasakian-space-form ${\overline{M}}^{2n+1}(f_1,f_2,f_3)$ is said to be $C$-totally real submanifold if $\xi \in \Gamma \left(T^{\perp }M\right)$ and $\phi X\in \Gamma \left(T^{\perp }M\right)$ for all $X\in \Gamma \left(TM\right)$. In particular, if $m=n$, then $M^n$ is called Legendrian submanifold. Here, we derive Wintgen inequalities on Legendrian submanifolds of generalized Sasakian-space-forms with respect to different connections; namely, quarter symmetric metric connection, Schouten-van Kampen connection and Tanaka-Webster connection.

The aim of this paper is to determine explicitly the algebraic structure of the curvature algebra of the 3-dimensional Heisenberg group with left invariant cubic metric. We show, that this curvature algebra is an infinite dimensional graded Lie subalgebra of the generalized Witt algebra of homogeneous vector fields generated by three elements.

We prove that Witten’s Conjecture [40] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_1=0$ and odd $b_2^{+}\ge 3$ follows from our $\left(3\right)$-monopole cobordism formula [6] when the four-manifold has $c_1^2\ge {\chi }_h-3$ or is abundant.