Related papers: Equiaffine isoparametric functions and their regul…

We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together…

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of…

In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs…

We introduce a nonstandard extension of the category of diffeological spaces, and demonstrate its application to the study of generalized functions. Just as diffeological spaces are defined as concrete sheaves on the site of Euclidean open…

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower…

In this paper, we address the following two general problems: given two algebraic varieties in ${\bf C}^n$, find out whether or not they are (1) isomorphic; (2) equivalent under an automorphism of ${\bf C}^n$. Although a complete solution…

Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the…

In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean \mathbb{R}^{n+1} with speed u^\alpha f^\beta (\alpha, \beta\in\mathbb{R}^1), where u is support function of the hypersurface, f is a…

We construct families of smooth affine surfaces with pairwise non isomorphic A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are…

We classify the isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$, $n, m\geq2$, with compact level sets, where $\mathbb{M}^m$ is a connected, closed Riemannian manifold of dimension $m$. Also, we classify the isoparametric…

We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly…

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that…

We investigate geometric invariants of cuspidal edges on focal surfaces of regular surface. In particular, we shall clarify the sign of the singular curvature at a cuspidal edge on a focal surface using singularities of parallel surface of…

Building on work of Furstenberg and Tzkoni, we introduce ${\bf r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the…

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean…

In this first work dedicated to the generalisation of classic algebraic geometry to non algebraically closed fields and axiomatisable classes of fields, we develop the foundations for equiresidual algebraic geometry (EQAG), i.e. algebraic…

Generative modeling seeks to uncover the underlying factors that give rise to observed data that can often be modeled as the natural symmetries that manifest themselves through invariances and equivariances to certain transformation laws.…

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the…

In this paper, we generalize the idea of equiaffine structure to the case of frontals and we define the Blaschke vector field of a frontal. We also investigate some necessary and sufficient conditions that a frontal needs to satisfy to have…