On the geometry of null hypersurfaces in Minkowski space

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space ${\mathbb{R}}_1^{n+2}$, classifying all rotation null hypersurfaces in ${\mathbb{R}}_1^{n+2}$. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the $(n+1)$-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the $3$-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of ${\mathbb{R}}_1^4$ and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.     

Monopoles on the Bryant–Salamon G2-manifolds

$G_2$-monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant–Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the ${\mathbb{S}}^4$ and ${\mathbb{C}\mathbb{P}}^2$. These are the first nontrivial examples of $G_2$-monopoles.Associated with each monopole there is a parameter $m\in {\mathbb{R}}^{+}$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m\in {\mathbb{R}}^{+}$ there is a unique monopole with mass $m$. We also find explicit irreducible $G_2$-instantons on ${\Lambda }_{-}^2\left({\mathbb{S}}^4\right)$ and on ${\Lambda }_{-}^2\left({\mathbb{C}\mathbb{P}}^2\right)$.The third Bryant–Salamon $G_2$-metric lives on the spinor bundle over the $3$-sphere. In this case we produce a vanishing theorem for monopoles.     

Chern–Simons formulation of three-dimensional gravity with torsion and nonmetricity

We consider various models of three-dimensional gravity with torsion or nonmetricity (metric affine gravity), and show that they can be written as Chern–Simons theories with suitable gauge groups. Using the groups $\mathrm{ISO}(2,1)$, $\mathrm{SL}(2,\mathbb{C})$ and $\mathrm{SL}(2,\mathbb{R})\times \mathrm{SL}(2,\mathbb{R})$, and the fact that they admit two independent coupling constants, we obtain the Mielke–Baekler model for zero, positive and negative effective cosmological constant respectively. Choosing $\mathrm{SO}(3,2)$ as the gauge group, one gets a generalization of conformal gravity that has zero torsion and only the trace part of the nonmetricity. This characterizes a Weyl structure. Finally, we present a new topological model of metric affine gravity in three dimensions arising from an $\mathrm{SL}(4,\mathbb{R})$ Chern–Simons theory.     

Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti-commuting for real hypersurfaces in complex two-plane Grassmannians $G_2\left({\mathbb{C}}^{m+2}\right)$ and prove a complete classification theorem, which gives a shrinking Ricci soliton with potential Reeb flow on Hopf real hypersurfaces and a tube over a totally real totally geodesic $\mathbb{Q}{P}^n$, $m=2n$ in $G_2\left({\mathbb{C}}^{m+2}\right)$.     

A new characterization of the Berger sphere in complex projective space

We give a complete classification of Lagrangian immersions of homogeneous 3-manifolds (the Berger spheres, the Heisenberg group ${\text{Nil}}_3$, the universal covering of the Lie group $\text{PSL}(2,\mathbb{R})$ and the Lie group ${\text{Sol}}_3$) in 3-dimensional complex space forms. As a corollary, we get a new characterization of the Berger sphere in complex projective space.     

The geometry of Π-invertible sheaves

Using the fact that $\Pi$-invertible sheaves can be interpreted as locally free sheaves of modules for the super skew field $\mathbb{D}$, we give a new construction of the $\Pi$-projective superspace ${\mathbb{P}}_{\Pi ,B}^n$ over affine $k$ superschemes $B$, $k$ an algebraically closed field. We characterize morphisms into ${\mathbb{P}}_{\Pi ,B}^n$ and give a new interpretation of the composition of $\Pi$-invertible sheaves in terms of the algebra of $\mathbb{D}$.