Bogdan D. Suceava

California State University, Fullerton | CSUF · Department of Mathematics

We recall several pieces of information about Professor Bang-Yen Chen’s life, works, as well as the lasting impact of his creative outcome. Our essay could prove useful to anyone interested in the history of differential geometry in the recent six decades. With over 36,000 citations (as presently recorded by ResearchGate), the body of work created...

By J.F. Nash’s Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash’s Theorem effectively is finding useful relationships between intrinsic and extrinsic elements that are characterizing immersions. After 1993...

By J.F. Nash’s Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large. S.-S. Chern pointed out in 1968 that a key technical element in applying Nash’s Theorem effectively is finding useful relationships between intrinsic and extrinsic elements that are characterizing immersions. After 1993,...

The present paradigm associates the dawn of modern applied mathematics with the first decades of the 19th century. In an investigation of these historical premises, we search for themes investigated today through methods pertaining to applied mathematics in the works of a medieval scholar whose singular vision helped him reach several conclusions t...

The main purpose of this chapter is to provide a detailed survey of recent results in complex slant geometry of Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds.

Professor Tadashi Nagano (1930-2017) was one of the great Japanese differential geometers, whose fundamental and seminal work still attracts much interest today. This volume, consisting of 13 contributed chapters, is inspired by his work and his legacy and, while reminding historical results obtained in the past, presents recent developments in the...

"What should we use?" seems to be the question when one approaches a plane geometry problem. In many ways, Euclidean geometry is a laboratory in the realm of logic, an ideal place where one can see how alternative methods can be employed to solve problems. What detail might represent a hint? And from among many choices, what method could one consid...

We are investigating a class of integrable functions which admit an integration process through elementary integration steps. We are using a class of natural $u$-substitutions given by linear bijections symmetrizing the interval of integration.

We inquire whether there are some fundamental interpretations of elementary inequalities in terms of curvature of a three-dimensional smooth hypersurface in the four-dimensional real ambient space. The main outcome of our exploration is a perspective of regarding the natural substance of some mathematical inequalities, which represent important phy...

This is a book review for a volume recently published by AMS’ Student Math Library series

John F. Nash Jr.'s Embedding Theorem, published originally in 1956, states that every Riemannian manifold can be isometrically embedded into some Euclidean space. This fundamental result is a very beautiful and extremely important result in differential geometry, and especially in the geometry of submanifolds. One of the researchers with outstandin...

The Nine-Point Circle Theorem was first obtained in a joint work written by C.-J. Brianchon and J.-V. Poncelet. Since the authors of this Theorem presented it in their original work as a principle in geometry, we are invited to reflect whether there are some direct applications of this Theorem in advanced Euclidean geometry. Our exploration only co...

The Nine-Point Circle Theorem was first obtained in a joint work written by C.-J. Brianchon and J.-V. Poncelet. Since the authors of this Theorem presented it in their original work as a principle in geometry, we are invited to reflect whether there are some direct applications of this Theorem in advanced Euclidean geometry. Our exploration only co...

This article appears in Didactica matematică, a Supplement of Gazeta matematică. In the fall of 1983, the Romanian bookstores distributed a 290-pages small format paperback produced by Albatros Press, titled The Charm of the Old-Fashioned Geometry (in original: Vraja geometriei demodate). It sold rather quickly an extensive printing, although it wa...

Sophie Germain was a reputed female mathematician, an independent academia thinker, who lived in France during the first Empire and Bourbon Restauration. She had remarkable contributions to number theory and elasticity theory and she was the first to introduce the concept of mean curvature. She was a contemporary of Gauss, Lagrange, Cauchy and Ponc...

This essay provides a brief sketch of selected mathematical research work of Bang-Yen Chen done during the last fifty years.

Linear Weingarten surfaces in three-dimensional ambient space satisfy a relation between mean curvature and Gaussian curvature: aH^2 +bK = c. We investigate whether for coreciprocal points on smooth strictly convex hy-persurfaces of dimensions 3 and 4 there are any curvature inequalities inspired by the classical Weingarten condition. We also consi...

This Letter to the Editor represents an homage to Dr. Paul Yiu, after he communicated that the editorial project 'Forum Geometricorum' will not accept any longer further submissions.

These are the slides prepared for a presentation in the Fullerton Mathematical Circle, an outreach program of the Department of Mathematics at Cal State Fullerton. The presentation describes the ideas related to Ptolemy's Theorem, from its incorporation into the Almagest, to the study of Ptolemaic metrics in contemporary metric geometry.

We are describing the historical context in which Kentaro Yano prepared his doctoral dissertation under Elie Cartan’s coordination, and how this work was published in Romania, with Analele Ştiinţifice ale Universităţii “Al. I. Cuza”. We describe some of the many encounters made possible by Élie Cartan’s extraordinary creative contributions, which l...

This article is written for the Anniversary Issue of the Bucharest-based monthly Gazeta matematică, celebrating 125 years from its first issue. We describe how for 10 years (2011-2020) we translated and worked the problems published every month in the Gazette with many interested middle- and high-school students from Orange County, California, in t...

B.-Y. Chen's δ^-invariants can be estimated in function of other curvature terms through an algebraic process using the AM-GM and AM-QM inequalities. This procedure works on strictly convex smooth hypersurfaces lying in an Euclidean ambient space, and the estimates relate some δ^-invariants to Germain's mean curvature and Casorati curvature. As a c...

We show that a curvature condition on the Gauss-Kronecker curvature and scalar curvature of a convex smooth hypersurface lying in the four dimensional Euclidean space yields a lower bound for the mean curvature. The curvature condition we investigate is suggested by the local geometry of cylinders in the four dimensional Euclidean space.

"It is a very appealing idea to treat only the special case of a result, and one for which students are often chastised. It is especially satisfying, then, to see how one can get away with such a transgression. We will do precisely this by proving geometric facts using affine transformations. In fact, there will be a two-fold benefit to these explo...

In 1932, David Hilbert and Stefan Cohn-Vossen published their inspiring book titled Geometry and the Imagination. This influential volume was preceded historically by the works of Alfred Clebsch and Felix Klein, in the second half of the 19th century. Since then, the quest for the profound meaning behind geometric ideas has a model, very difficult...

It is of major interest to point out natural connections between the geometry of triangles and various other areas of mathematics. In the present work we show how Euler’s classical inequality between circumradius and inradius inspires, by using a duality between triangle geometry and three-dimensional hypersurfaces lying in the four dimensional Euc...

In a visionary short paper published in 1855, Ossian Bonnet derived a theorem relating prescribed curvature conditions to the admissible maximal length of geodesics on a surface. Bonnet’s work opened the pathway for the quest of further connections between curvature conditions and other geometric properties of surfaces, hypersurfaces or Riemannian...

Isidore’s 'Etymologies' enjoyed a wide audience during the medieval period. We investigate the structure of mathematics, as it is described in the 'Etymologies', and we discuss the sources on which Isidore relied when he described the structure of mathematics. A change of paradigm took place in Europe after the Recovery of Aristotle, in later centu...

In 1968, S.-S. Chern raised to attention the question that there might be further Riemannian obstructions for a manifold to admit an isometric minimal immersion into an Euclidean space. This question triggered a whole new direction of study in the geometry of submanifolds. ‪In the last two decades, there have been important advances in the study of...

The geometry of three-dimensional hypersurfaces in the four-dimensional Euclidean space enjoys certain interesting properties. We highlight some of these properties and indicate in our presentation a few references where this work is discussed.

It is a paradox that, after the catastrophic CE fifth century, notorious for the great invasions of the Western Europe and the Mediterranean Basin by populations coming from different geographical areas and the ultimate fall of the Roman Empire, the Roman culture continued to develop and several Latin works were produced. A natural question from th...

The concept of curvature appears for the first time in the monograph titled De configurationibus, written by Nicole Oresme around 1350. We describe this contribution and its historical context, as well as further implications of Oresme’s “doctrine of configurations”. Additionally, we plan to explore a few recent interpretations of the idea of curva...

We investigate whether a series with terms defined recursively by the relation xn+1 = f (xn), for all n ≥ 1, converges or diverges. For the examples studied, the ratio test is inconclusive. We provide a theorem to decide whether such series are either convergent or divergent under some natural analytic conditions.

We explore the connection between the geometries generated by logarithmic oscillations and the class of metric spaces satisfying the Gromov hyperbolicity condition. We investigate the most fundamental examples, inspired from classical geometries, e.g. the Euclidean distance on the infinite strip or Hilbert’s distance on the unit disk. We continue o...

This poster was presented by Evelyn R. Easdale at the Spring 2016 MAA Meeting Southern California Nevada. It was recognized as an MAA Outstanding Poster. The content of the paper written on this topic was published in the American Mathematical Monthly one year later.

In the present work, we focus on how the geometries determined by metrics defined through logarithmic oscillations, developed originally in Barbilian’s work, are naturally connected to the current study of Gromov hyperbolic spaces. We introduce a distance on a subset of the n-dimensional real space and we call it stabilizing metric. We show it is a...

Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds in a Riemannian manifold (1968)], an important study of the very idea of curvature was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades, B.-Y. Chen...

Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds in a Riemannian manifold (1968)], an important study of the very idea of curvature was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades, B.-Y. Chen...

The amalgamatic curvature A(p) is a natural geometric quantity whose construction parallels that of classical scalar curvature. Its role in a ladder of curvatures corresponds to the role of harmonic mean in the classical ladder of power means, i.e. to the mean of power −1. In the present work we determine lower and upper bounds for the range of the...

An elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C . d; where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid's axis. This rigidity property was discovered by M. Chasles in the first half of the XIXth century....

An elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C • d, where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid's axis. This rigidity property was discovered by M. Chasles in the first half of the XlXth century....

An elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C · d, where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid’s axis. This rigidity property was discovered by M. Chasles in the first half of the XIXth century....

We pose and investigate the inverse curvature problem. That is, we explore when elements of a certain family of monomials represent the curvature function of a curve specified by combinations of elementary functions. The main result utilizes an integration theorem of Chebyshev.

The description of the course of Foundations of geometry presented by the author for the upper division undergraduate students at California State University, Fullerton. The course combines G. Venema's texbook 'Foundations of Geometry' with the study of several historically important papers, e.g. L. Euler's paper from 1767, or Brianchon and Poncele...

Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of intrinsic Riemannian obstructions to minimality, an important study of the very idea of curvature was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades, B.-Y. Chen's fundamental inequalities have been investigated by many authors in...

We introduce a string of new curvature invariants of a hypersurface in the real (n+1)-dimensional space and we establish a ladder of inequalities involving these curvature invariants. There is an analogy between this series of inequalities and the classical ladder of power means for positive real numbers. To describe the natural geometric interpret...

We give two alternate proofs of B.-Y. Chen's inequality with classical curvature invariants for 3-dimensional hypersurfaces in E4 . Then, by using an idea described by Cvetkovski, we obtain an estimate of the Gauss-Kronecker curvature of a three-dimensional smooth hypersurface in the four dimensional Euclidean space in function of its mean curvatur...

The spread of a matrix is introduced by Mirsky in 1956 in [20]. The classical theory provides an upper bound for the spread of the shape operator in terms of the second fundamental form of a hypersurface in the Euclidean space. In the present work, we are extending our understanding of the phenomenon by proving a lower bound, inspired from an idea...

U. Dursun obtained explicit parametrizations of rotation hyper- surfaces in the Lorentz-Minkowski ambient space. The pointwise umbilicity condition yields a di�erential equation in each of the cases described by Dur- sun's parametrizations. In the present work we study the direct solutions of these di�erential equations.

In the most recent decades, metric spaces have been studied from a variety of viewpoints. One of the important characterizations developed in the study of distances is Gromov hyperbolicity. Our goal here is to pro- vide two approachable, but also intuitive examples of Gromov hyperbolic metric spaces. The authors believe that such examples could be...

In the present work we prove that one of Barbilianʼs theorems from 1960 regarding the metrization procedure in the plane admits a natural extension depending on a bilinear form and the relative position of two Apollonian hyperspheres. This result allows us to pursue two fundamental ideas. First, that all the distances with constant curvature can be...

Surfaces in 4D Riemannian space forms have been investigated extensively. In contrast, only few results are known for surfaces in 4D neutral indefinite space forms R^4_2(c). Thus, in this paper we study space-like surfaces in R^4_2(c) satisfying certain simple geometric properties. In particular, we classify space-like surfaces in E^4_2 with consta...

Introduced originally in 1934, Barbilian’s metrization procedure induced a distance on a planar domain by a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to domains of a more general form, withstanding not necessarily on planar sets, but in a more abstract setting. In the present work, we...

We prove that the surface in three dimensional Euclidean space generated by the formula of the Euler line in a triangle satisfi�es Tzitzeica's a�ffine invariance property. This result establishes an interesting connection between triangle geometry and the property that lies at the origins of a�ffine differential geometry.

We describe the historical and ideological context that brought to the fore the study of a centro-affine invariant that subsequently received much attention. The invariant was introduced by Ţiţeica in 1907, and this discovery has been viewed by many as a consequence of Klein's Erlangen program. We thus present the starting point of affine different...

93.15 Some theorems about perpendicular lines, proved using an extension of Pythagoras’ theorem - Volume 93 Issue 526 - Wladimir G. Boskoff, Laurenţiu Homentcovschi, Bogdan D. Suceavă

In the present paper we answer two questions raised by Barbilian in 1960. First, we study how far can the hypothesis of Barbilian’s metrization procedure can be relaxed. Then, we prove that Barbilian’s metrization procedure in the plane generates either Riemannian metrics or Lagrance generalized metrics not reducible to Finslerian or Langrangian me...

According to various sources (e.g. [1, p. 102]), the terminology of the power of a point with respect to a circle is due to Steiner. His definition appears in most classical and contemporary geometry textbooks (to mention just a few references, see [2, 3, 4, 5]). The concept of the power of a point has been revisited not only in advanced Euclidean...

In the present note, we survey the most recent results related to Chen's fundamental inequality with classical invariants, originally proved in [12] and also studied by the �rst author in [28]. In the last section, we apply Chen's fundamental inequality with classical invariants to warped products. This study is done in the context of B.-Y. Chen's...

In this note we obtain a new cyclicity criterion for four points in the Euclidean plane, by using algebraic and geometric structures induced in ℂ2 by the two dimensional complex projective space. We show that if four points lie on a circle in the real plane, then the type-one isotropic lines intersect z2-axis in four points of real cross ratio.

Barbilian spaces are metric spaces with a metric induced by a special procedure of metrization that is inspired by the study of the models of non-Euclidean geometry. In this note we discuss the history of Barbilian spaces and the evolution of the theory. We point out that some of the current references to work done in Barbilian spaces refer to Barb...

In this note we discuss a geometric viewpoint on Rolle's Theorem and we show that a particular setting of the form of Rolle's Theorem yields a metric that is the hyperbolic metric on the disk. Our result is related to recent developments in the study of Barbilian's metrization procedure.

Barbilian spaces are metric spaces with a metric induced by a special procedure of metrization which is inspired by the study of the models of non-Euclidean geometry. In the present material we discuss the history of Barbilian spaces and the evolution of the theory. We point out that some of the current references to the work done in Barbilian spac...

Several authors have pointed out the connection between Barbilian's metric introduced in 1934 and the recent study of Apollonian metrics. We provide examples of various distances that can be obtained by Barbilian's metrization procedure and we discuss the relation between this metrization procedure and important Riemannian and generalized Lagrangia...

Discussing a series of inequalities, B. Bollobas reminds us in (3) that Harald Bohr wrote: "All analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove." Fortunately, other inequalities can be reduced to techniques whose strategy of proof is familiar to us. This expository note has been...

We study an interesting configuration that gives an example of an elliptic projectivity characterized by the Pythagorean relation.

We study a class of geometric identities and inequalities that have a common pattern: they are generated by a homogeneous function. We show how to extend some of these homogeneous relations in the geometry of triangle. Then, we study the geometric configuration created by two intersecting lines and a pencil of n lines, where the repeated use of Men...