Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Möbius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {E_i}. Then we give a complete classification for spacelike Möbius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Möbius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Möbius form.     

Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Möbius addition is defined on the complex open unit disk by

$$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$

and Möbius’s exponential equation takes the form \(L(a\oplus _M b) = L(a)L(b)\), where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius’s exponential equation is connected to Cauchy’s exponential equation. Möbius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.

     

On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

The aim of this paper is twofold. On the one hand, we enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau–Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter, respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow us to prove the minimum principle and one version of the open mapping theorem. On the other hand, we adopt a completely new approach to strengthen a version of boundary Schwarz lemma first proved in Ren and Wang (Trans Am Math Soc 369:861–885, 2017) for quaternionic slice regular functions. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.     

Möbius and triangle semigroups

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (?, SL(2,?)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius semigroup.     

$$\partial \overline{\partial }$$-Complex symplectic and Calabi–Yau manifolds: Albanese map,deformations and period maps

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms.     

A complete classification of Blaschke parallel submanifolds with vanishing Mbius form

The Blaschke tensor and the Mbius form are two of the fundamental invariants in the Mobius geometry of submanifolds;an umbilic-free immersed submanifold in real space forms is called Blaschke parallel if its Blaschke tensor is parallel.We prove a theorem which,together with the known classification result for Mobius isotropic submanifolds,successfully establishes a complete classification of the Blaschke parallel submanifolds in S~n with vanishing Mobius form.Before doing so,a broad class of new examples of general codimensions is explicitly constructed.     

On Möbius form and Möbius isoparametric hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.     

Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their f-vectors and some additional constraints.     

The Möbius function of a composition poset

We determine the Möbius function of a poset of compositions of an integer. In fact, we give two proofs of this formula, one using an involution and one involving discrete Morse theory. This composition poset turns out to be intimately connected with subword order, whose Möbius function was determined by Björner. We show that, using a generalization of subword order, we can obtain both Björner’s results and our own as special cases.     

Catégories de Möbius et fonctorialités: un cadre général pour l'inversion de Möbius

This paper proposes a general setting for Möbius inversion which includes the cases of locally finite partially ordered sets and monoids with the finite decomposition property (f.d.p.). The unifying concept is that of categories with the f.d.p., called Möbius categories, which we characterize in four different ways. The established theory of incidence algebras, Möbius functions, product formulas, reduced algebras, etc., is carried over. Furthermore, the study of functors between Möbius categories yields results in two directions: firstly, inspired from the substitution of formal power series is the construction of a homomorphism between incidence algebras which allows the transfer of Möbius inversions; secondly, surjective functors often give rise to a reduced incidence algebra, thus shedding new light on many important reduced algebras in combinatorial theory.     

Möbius homogeneous hypersurfaces with two distinct principal curvatures in S n+1

The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in S ⁿ⁺¹ under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in S ⁴.     

Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

     

The Möbius function of permutations with an indecomposable lower bound

We show that the Möbius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the Möbius sum. For increasing oscillations, we give a recursion for the Möbius sum which only involves evaluating simple inequalities.     

Möbius inversion formula for monoids with zero

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the Möbius functions of a monoid and its Rees quotient are also provided.     

The foundations of enumeration theory for finite nilpotent groups

This paper is the first in a series of papers that lay the foundations of enumeration theory for finite groups including the classical inversion calculus on segments of the natural series and on lattices of subsets of finite sets. Since it became possible to calculate the Möbius function on all subgroups of finite nilpotent groups, the Möbius inversion on these groups began to play a decisive role. The efficiency of the inversion method as a regular technique suitable for solution of enumeration problems of group theory is illustrated with a number of concrete and very important enumerations. Bibliography: 13 titles.     

On the Möbius Function of a Pointed Graded Lattice

In this paper, we compute the Möbius function of pointed integer partition and pointed ordered set partition using topological and analytic methods. We show that the associated order complex is a wedge of spheres and compute the associated reduced homology group for each subposet. In addition, we compute the Möbius function of pointed graded lattice and use our method to compute the Möbius function of pointed direct sum decomposition of vector spaces.     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.     

Möbius characterization of the boundary at infinity of rank one symmetric spaces

Möbius structure (on a set \(X\) ) is a class of metrics having the same cross-ratios. A Möbius structure is Ptolemaic if it is invariant under inversion operations. The boundary at infinity of a \(\mathrm{CAT }(-1)\) space is in a natural way a Möbius space, which is Ptolemaic. We give a free of classification proof of the following result that characterizes the rank one symmetric spaces of noncompact type purely in terms of their Möbius geometry: Let \(X\) be a compact Ptolemy space which contains a Ptolemy circle and allows many space inversions. Then \(X\) is Möbius equivalent to the boundary at infinity of a rank one symmetric space.     

Möbius geometry of three-dimensional Wintgen ideal submanifolds in \mathbb{S}^5

Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the socalled DDVV inequality which relates the scalar curvature,the mean curvature and the normal scalar curvature.This property is conformal invariant;hence we study them in the framework of Mbius geometry,and restrict to three-dimensional Wintgen ideal submanifolds in S5.In particular,we give Mbius characterizations for minimal ones among them,which are also known as(3-dimensional)austere submanifolds(in 5-dimensional space forms).