Geometric Structures on Manifolds: Riemannian Geometry, Submanifolds, and Physical Applications

Topic Information

Dear Colleagues,

The Topic is devoted to Riemannian manifolds and submanifolds equipped with various additional geometric structures, such as foliation, distribution, almost product, almost contact, and complex structures. The topics covered in this Topic include the following: local and global differential geometry, curvature and topology of Riemannian manifolds and submanifolds (regular and with singularities); geometric inequalities related to extrinsic to intrinsic curvature invariants; variational problems for curvature functionals; almost contact (Sasakian, Kenmotsu, cosymplectic, etc.) structures and Ricci-type solitons; manifolds with density; etc. Particular attention is also given to geometric structures arising in mathematical physics, for example, non-symmetrical gravitational theory, including special affine connections with torsion and non-symmetric metricity conditions, as well as their connection with almost Hermitian and almost contact structures.

Prof. Dr. Vladimir Rovenski
Prof. Dr. Milan Lj Zlatanović
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Axioms axioms	1.5	-	2012	21.6 Days	CHF 2400	Submit
Foundations foundations	-	-	2021	24.3 Days	CHF 1000	Submit
Geometry geometry	-	-	2024	15.0 days *	CHF 1000	Submit
Mathematics mathematics	2.3	5.4	2013	17.4 Days	CHF 2600	Submit
Symmetry symmetry	2.2	5.2	2009	16.3 Days	CHF 2400	Submit

* Median value for all MDPI journals in the first half of 2026.

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Published Papers (3 papers)

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All Axioms Geometry Mathematics Symmetry Foundations

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21 pages, 297 KB  

Open AccessArticle

The Evolution of Mathematization: From Classical Science to Computationally Emergent Structures

by Cécile Barbachoux

Foundations 2026, 6(3), 27; https://doi.org/10.3390/foundations6030027 - 16 Jul 2026

Viewed by 299

Abstract

The mathematization of science is undergoing a structural transformation driven by the rise of computation and data-intensive methods. While classical mathematization relied largely on explicitly formulated laws and formal structures, contemporary scientific practice increasingly encounters mathematical objects that arise from dynamical and algorithmic [...] Read more.

The mathematization of science is undergoing a structural transformation driven by the rise of computation and data-intensive methods. While classical mathematization relied largely on explicitly formulated laws and formal structures, contemporary scientific practice increasingly encounters mathematical objects that arise from dynamical and algorithmic processes. This paper introduces the notion of computationally emergent structures to describe objects, representations, or effective functional spaces that are generated and stabilized through the interaction of parameterized models, optimization dynamics, and data. We propose a minimal formal schema in which such structures can be understood as stabilized outcomes of learning dynamics. In overparameterized regimes, this schema clarifies how optimization procedures may select particular solutions through implicit biases or variational tendencies that are not specified a priori. framework brings together implicit regularization, kernel regimes, and stability phenomena in modern learning systems, while distinguishing between established formal results and broader epistemological interpretation. It suggests that contemporary learning systems provide a privileged setting in which geometry, dynamics, and data jointly contribute to the production of effective mathematical structure. This perspective identifies a shift from representation to dynamical emergence and extends the study of mathematization toward an analysis of structure formation grounded in computation. Full article

(This article belongs to the Topic Geometric Structures on Manifolds: Riemannian Geometry, Submanifolds, and Physical Applications)

14 pages, 913 KB  

Open AccessFeature PaperArticle

Information Geometry Description of Inferential Scattering

by Marco Favretti

Geometry 2026, 3(3), 12; https://doi.org/10.3390/geometry3030012 - 30 Jun 2026

Viewed by 357

Abstract

We investigate the geometrical structure underlying the notion of inferential scattering, which was formulated by E. T. Jaynes in the 1980s using the language of equilibrium statistical mechanics. We show that inferential scattering can be naturally defined on a dually flat Riemannian manifold [...] Read more.

We investigate the geometrical structure underlying the notion of inferential scattering, which was formulated by E. T. Jaynes in the 1980s using the language of equilibrium statistical mechanics. We show that inferential scattering can be naturally defined on a dually flat Riemannian manifold equipped with dual coordinate systems, a differential-geometric structure that occupies a central place in Information Geometry. We find that the controlled evolution of the system on the dually flat manifold can be expressed as the horizontal lift of an integrable connection. Full article

(This article belongs to the Topic Geometric Structures on Manifolds: Riemannian Geometry, Submanifolds, and Physical Applications)

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22 pages, 334 KB  

Open AccessArticle

Magnetic Curves in Generalized Almost Cosymplectic Manifolds

by Foued Aloui, Afshan Perween and Majid Ali Choudhary

Symmetry 2026, 18(5), 808; https://doi.org/10.3390/sym18050808 - 8 May 2026

Viewed by 376

Abstract

In this paper, we study magnetic curves associated with a contact magnetic field on almost cosymplectic manifolds. In particular, we consider locally conformal almost cosymplectic manifolds and almost $\alpha$-cosymplectic f-manifolds. The magnetic field is defined by the fundamental 2-form $\Omega$, [...] Read more.

In this paper, we study magnetic curves associated with a contact magnetic field on almost cosymplectic manifolds. In particular, we consider locally conformal almost cosymplectic manifolds and almost $\alpha$-cosymplectic f-manifolds. The magnetic field is defined by the fundamental 2-form $\Omega$, with the corresponding Lorentz force determined by the structure tensor $\phi$. We classify normal magnetic curves in these settings and show that they are either geodesics, Legendre $\phi$-circles, or $\phi$-helices of order three. We further investigate slant curves and derive conditions relating their geometry to the structure functions of the manifold. These results generalize known classifications of magnetic curves in contact and cosymplectic geometries and provide a unified treatment for the considered classes of manifolds. Full article

(This article belongs to the Topic Geometric Structures on Manifolds: Riemannian Geometry, Submanifolds, and Physical Applications)

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