Ibrahim, Fatima and Al-obaidi, Shaymaa S.A. (2026) Geometric–Topological Interplay in Modern Riemannian Geometry: Extensions and Applications of the Chern–Gauss–Bonnet Theorem. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 7 (4). ISSN 2660-5309

Text CAJMTCS_Geometric–Topological.pdf Download (509kB)

Abstract

The aim of the present study is to investigate the geometric–topological connection between curvature and global topology with emphasis on the mathematical foundations, extensions and applications to modern Riemannian geometry from the Gauss–Bonnet theorem and the Chern–Gauss–Bonnet theorem. The approach of the research is analytical and theoretical method, with the use of differential geometric technique, differential form, and mathematical derivation. First, with the aid of triangulation, the classical Gauss–Bonnet theorem is proved for compact, oriented two-dimensional surfaces with boundary, showing how the interior geodesic curvature contributions cancel, and how the vertex angle sums determine the Euler characteristic. Next, the higher dimensional formulation is studied with the help of the Cartan structure equations, the curvature forms and the Euler class in Pfaffian representation. Special emphasis is placed on Chern's intrinsic proof including the proof of the existence of the transgression form on the unit tangent sphere bundle and the fundamental differential equation for the transgression form. The study also contains remarks on boundary extensions, Euler characteristic and the Poincaré–Hopf and the Atiyah–Singer index theorems. Explicit evaluations in the two-dimensional and four-dimensional spheres give examples of the relation between curvature integrals and the topological invariants. Moreover, the investigation discusses the five dimensional Einstein–Gauss–Bonnet gravitational framework in the context of the Bouleware–Deser black-hole solution. When the Gauss–Bonnet coupling becomes small, its physical branch is obtained, which is the five-dimensional Schwarzschild–Tangherlini metric of general relativity. In general, the results shed light on how the local curvature information influences the global topological properties and illustrate the significance of the Gauss–Bonnet invariant in higher dimensional geometric and gravitational models. Future research directions identified in the study involve extensions in Discrete Differential Geometry, to non-compact manifolds, and Singular Spacetime Geometries.

Item Type:	Article
Uncontrolled Keywords:	Differential Geometry, Gauss–Bonnet Theorem, Chern–Gauss–Bonnet Theorem, Riemannian Manifolds, Euler Characteristic, Pfaffian, Transgression Form, instein–Gauss–Bonnet Gravity, Boulware–Deser Solution
Subjects:	H Social Sciences
Depositing User:	admin eprints
Date Deposited:	22 Oct 2026 01:28
Last Modified:	22 Oct 2026 01:28
URI:	http://eprints.umsida.ac.id/id/eprint/17230

Actions (login required)

View Item