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Search Results (3)

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Keywords = de Rham cohomology groups

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28 pages, 999 KiB  
Article
Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields
by Peter D. Neilson and Megan D. Neilson
AppliedMath 2024, 4(2), 763-790; https://doi.org/10.3390/appliedmath4020041 - 15 Jun 2024
Viewed by 1313
Abstract
At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d [...] Read more.
At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., ω=dλ where λ is an r−1-form) is closed (i.e., dω=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains. Full article
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13 pages, 273 KiB  
Article
2D Discrete Hodge–Dirac Operator on the Torus
by Volodymyr Sushch
Symmetry 2022, 14(8), 1556; https://doi.org/10.3390/sym14081556 - 28 Jul 2022
Cited by 1 | Viewed by 1784
Abstract
We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide [...] Read more.
We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups. Full article
(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)
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11 pages, 300 KiB  
Article
Topological and Geometrical Properties of k-Symplectic Structures
by Essabab Said, Fanich El Mokhtar and Awane Azzouz
Axioms 2022, 11(6), 273; https://doi.org/10.3390/axioms11060273 - 6 Jun 2022
Cited by 1 | Viewed by 1975
Abstract
We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polarized k-symplectic [...] Read more.
We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polarized k-symplectic manifolds in a way analogous to symplectic manifolds which are all orientable. Full article
(This article belongs to the Collection Topological Groups)
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