Special Issue "Differential Geometry: Theory and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebraic Geometry".

Deadline for manuscript submissions: 30 November 2020.

Special Issue Editor

Prof. Juan De Dios Pérez
Website
Guest Editor
Departamento de Geometria y Topologia Facultad de Ciencias, Universidad de Granada E-18071 Granada, Spain
Interests: differential geometry; Riemannian geometry

Special Issue Information

Dear Colleagues,

Differential geometry can be considered to have been born in the middle of the 19th century, and from this moment, it has had several applications not only in mathematics, but in many other sciences. One can think, for example, about applications of the theory of curves and surfaces in the Euclidean plane and space. Differential geometry can be defined as the study of the geometry of differential manifolds, as well as of their submanifolds, and when these spaces are equipped with a metric (not necessarily Euclidean), one arrives at pseudo-Riemannian geometry and the main tool of curvature of a manifold, a concept with fundamental applications in physics, for instance, in the study of spacetimes.

In addition, applications of differential geometry can be found in almost any field of science, form biology to architecture.

This Special Issue is intended to provide a series of papers focused on the study of the problems in differential geometry, such as the different structures that one can consider on a differentiable or (pseudo) Riemannian manifold and its submanifolds, such as vector fields, forms, different kinds of tensor fields, fiber bundles, affine connections on manifolds, and how to apply them to other fields of science.

Prof. Juan De Dios Pérez
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Differentiable manifolds
  • (pseudo) Riemannian geometry
  • Submanifolds
  • Spacetimes
  • Physics
  • Statistics
  • Curvature
  • Fiber bundles
  • Invariants
  • Contact structures
  • Other sciences

Published Papers (6 papers)

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Research

Open AccessArticle
Generalized Sasakian Space Forms Which Are Realized as Real Hypersurfaces in Complex Space Forms
Mathematics 2020, 8(6), 873; https://doi.org/10.3390/math8060873 (registering DOI) - 29 May 2020
Abstract
We prove a classification theorem of the generalized Sasakian space forms which are realized as real hypersurfaces in complex space forms. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
Open AccessArticle
Singular Special Curves in 3-Space Forms
Mathematics 2020, 8(5), 846; https://doi.org/10.3390/math8050846 - 23 May 2020
Abstract
We study the singular Bertrand curves and Mannheim curves in the 3-dimensional space forms. We introduce the geometrical properties of such special curves. Moreover, we get the relationships between singularities of original curves and torsions of another mate curves. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
Open AccessArticle
A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms
Mathematics 2020, 8(4), 642; https://doi.org/10.3390/math8040642 - 21 Apr 2020
Abstract
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k [...] Read more.
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
Open AccessArticle
A New Angular Measurement in Minkowski 3-Space
Mathematics 2020, 8(1), 56; https://doi.org/10.3390/math8010056 - 02 Jan 2020
Abstract
In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and [...] Read more.
In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
Open AccessFeature PaperArticle
The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space
Mathematics 2019, 7(12), 1211; https://doi.org/10.3390/math7121211 - 09 Dec 2019
Abstract
We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the [...] Read more.
We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
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Open AccessArticle
Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures
Mathematics 2019, 7(12), 1162; https://doi.org/10.3390/math7121162 - 02 Dec 2019
Abstract
We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that [...] Read more.
We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that measure on the unit interval. We also investigate the normalized/symmetric conditions for operator-monotone functions. These conditions turn out to characterize monotone metrics and Morozowa–Chentsov functions as well. Concrete integral representations of such functions related to well-known monotone metrics are also provided. Moreover, we use this integral representation to decompose positive operator-monotone functions. Such decomposition gives rise to a decomposition of the associated monotone metric. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
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