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Mathematics
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Analysis on Differentiable Manifolds
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Analysis on Differentiable Manifolds

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "B: Geometry and Topology".

Deadline for manuscript submissions: 20 August 2025 | Viewed by 556

Special Issue Editor

Dr. Adara M. Blaga

E-Mail Website
Guest Editor
Department of Mathematics, West University of Timişoara, 300223 Timişoara, Romania
Interests: Ricci-Bourguignon solitons; statistical manifolds; polynomial structures and affine connections in generalized geometry; warped product and slant submanifolds; magnetic and biharmonic curves and surfaces; multisymplectic structures
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

An important problem in submanifold theory consists of finding optimal inequalities which relate extrinsic to intrinsic invariants for submanifolds in different ambient manifolds (endowed with polynomial structures and affine connections). Such invariants and inequalities have many applications in several areas of mathematics and related sciences. Minimal surfaces are known to be good mathematical models for physical phenomena, having been intensively studied in general relativity (black holes), cell biology (the endoplasmic reticulum), soap films, and materials science.

Geometric flows on (pseudo-)Riemannian manifolds are usually associated with extrinsic or intrinsic curvatures. One of the most studied flows is the Ricci flow introduced by Hamilton, whose self-similar solutions are Ricci solitons, which constitute natural generalizations of Einstein metrics. Another generalization of Einstein manifolds is the quasi-Einstein manifolds, important in the general theory of relativity, e.g., Robertson–Walker spacetime.

This Special Issue aims to collect reviews or research papers on topics concerning the geometry and topology of manifolds and submanifolds, as well as applications in mathematics or other scientific areas. Such topics can include, but are not limited to the following: manifolds with polynomial structures, (pseudo-)Riemannian metrics, and affine connections; distributions, foliations, and submanifolds; distinguished vector fields, vector bundles, and fiber bundles; geometric flows and solitons; spacetimes, etc.

Dr. Adara M. Blaga
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 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 manifold
  • (pseudo-)Riemannian metric
  • submanifold
  • curvature
  • optimal inequalities
  • affine connection
  • polynomial structure
  • quasi-Einstein manifold
  • spacetime
  • geometric flow
  • soliton
  • vector field
  • fiber bundle
  • vector distribution
  • foliations

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Published Papers (3 papers)

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Research

12 pages, 253 KiB  
Article
Pinching Results on Totally Real Submanifolds of a Locally Conformal Kähler Manifolds
by Noura M. Al-Houiti, Ali H. Alkhaldi, Akram Ali and Piscoran Laurian-Ioan
Mathematics 2025, 13(10), 1682; https://doi.org/10.3390/math13101682 (registering DOI) - 21 May 2025
Abstract
This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. [...] Read more.
This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. Full article
(This article belongs to the Special Issue Analysis on Differentiable Manifolds)
15 pages, 280 KiB  
Article
Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields
by Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri
Mathematics 2025, 13(10), 1672; https://doi.org/10.3390/math13101672 - 20 May 2025
Abstract
This paper investigates compact Riemannian hypersurfaces immersed in (n+1)-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined [...] Read more.
This paper investigates compact Riemannian hypersurfaces immersed in (n+1)-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space Rn+1, Euclidean sphere Sn+1, and de Sitter space S1n+1. Full article
(This article belongs to the Special Issue Analysis on Differentiable Manifolds)
33 pages, 458 KiB  
Article
Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics
by Bang-Yen Chen
Mathematics 2025, 13(9), 1417; https://doi.org/10.3390/math13091417 - 25 Apr 2025
Viewed by 332
Abstract
The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the [...] Read more.
The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e., Δ2x=0. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book. Full article
(This article belongs to the Special Issue Analysis on Differentiable Manifolds)
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