Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

Author:

Neilson Peter D.1ORCID,Neilson Megan D.2ORCID

Affiliation:

1. School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia

2. Independent Researcher, Late School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia

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 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.

Publisher

MDPI AG

Reference30 articles.

1. Neilson, P.D., Neilson, M.D., and Bye, R.T. (2018). A Riemannian geometry theory of three-dimensional binocular visual perception. Vision, 2.

2. Neilson, P.D., Neilson, M.D., and Bye, R.T. (2021). A Riemannian geometry theory of synergy selection for visually-guided movement. Vision, 5.

3. Neilson, P.D., Neilson, M.D., and Bye, R.T. (2022). The Riemannian geometry theory of visually-guided movement accounts for afterimage illusions and size constancy. Vision, 6.

4. Tasman, W., and Jaeger, E.A. (2006). The human eye as an optical system, Chapter 33. Duane’s Clinical Ophthalmology, Lippincott, Williams and Wilkins.

5. Armstrong, B., Aitken, J., Sim, M., and Swan, N. (2007). Final Report of the Independent Review and Scientific Investigation Panel, Australian Broadcasting Corporation.

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