Related papers: A Geometric Representation of Improper Indefinite …
We introduce an (equi-)affine invariant diffusion geometry by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to construct an invariant…
A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are…
An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense…
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
The covering of the affine symmetry group, a semidirect product of translations and special linear transformations, in $D \geq 3$ dimensional spacetime is considered. Infinite dimensional spinorial representations on states and fields are…
In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between…
The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of…
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of…
We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of…
Special generic maps are generalizations of Morse functions with exactly two singular points on spheres and canonical projections of unit spheres. They restrict the manifolds of the domains strongly in considerable cases and are important…
Every $\mathbb{A}^{1}-$bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of…
The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are…
The Sphere Covering Inequality was introduced in \cite{GM} (\emph{Invent. Math.}, 2018) as a sharp geometric inequality that provides a lower bound for the total area of two distinct surfaces of Gaussian curvature 1. These surfaces are…
We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical…
Deleting a hyperplane from a polar space associated with a symplectic polarity we get a specific, symplectic, affine polar space. Similar geometry, called an \afsempol\ arises as a result of generalization of the notion of an alternating…
If $c, \overline c\colon [a,b]\to \mathbb R^2$ are two convex planar curve parameterized by affine arc length and $A\colon [a,b]\to [0,\infty)$ is the area bounded by the restriction $c\big|_{[a,s]}$ and the segment between $c(a)$ and…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI),…
In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented…