Current Topics in Differential Geometry and Manifolds: Theory and Applications

Dear Colleagues,

The theory of manifolds was originally proposed by Riemannian as a generalization of surfaces. This allowed for the creation of geometry in the broadest possible range, encompassing both Euclidean and non-Euclidean geometries, and found application in many real-world problems. In particular, concepts such as manifolds and their associated definitions—metric tensors, Ricci tensors, and scalar curvature—which are central to Einstein's general theory of relativity, became key words in understanding this theory.

With nonlinearity finding more applications today, manifolds have become more important. Indeed, in AI, manifold techniques are used when data or model parameters naturally lie on non-Euclidean spaces. Using ordinary Euclidean geometry can distort these structures.

The following are possible uses for some fundamental concepts in manifold theory. Geodesics measures shortest paths on a curved surface. Tangent Space enables us to provide a local flat approximation where calculations are easier. Riemannian Optimization is gradient-based optimization that respects manifold constraints.

The above argument shows that Riemannian methods allow machine learning models to respect the true geometry of data, often leading to better representations, more efficient optimization, and improved performance on structured or constrained problems. Current research areas in machine learning where manifold theory is used include the following: for manifold learning, algorithms such as Isomap and locally linear embedding discover low-dimensional structures in data. Indeed, methods such as Principal Component Analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE), uniform manifold approximation and projection (UMAP) are used to simplify the high-dimensional manifolds. Geometric Deep Learning: Geometric Deep Learning extends neural networks to graphs and manifolds.

This Special Issue aims to present the latest research in the theory and applications of manifolds to the reader. This Special Issue will accept high-quality papers containing original research results and survey articles of exceptional merit in the following fields:

• Riemannian and semi-Riemannian Geometry;
• Generalized Geometries;
• Submanifold Theory (Including the theory of curves and surfaces);
• Differential Geometry and General Relativity;
• Riemannian submersions and related topics;
• Riemannian Geometry in Machine Learning;
• Riemannian maps and related topics;
• Inequalities on Riemannian manifolds;
• Computer Aided Geometric Design ( Bezier Curves and Surfaces, Spline curves and Surfaces).

Prof. Dr. Bayram Sahin
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 250 words) can be sent to the Editorial Office for assessment.

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-anonymized 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.

• Riemannian manifold
• semi-Riemannian manifold
• Riemannian submanifold
• manifold equipped with structure
• Riemannian manifold learning
• Bezier and Bezier-like curves and surfaces
• Riemannian submersion
• Riemannian map
• Ricci soliton