Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

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.

Keywords

• simplicial complexes
• singular homology groups
• singular cohomology groups
• singular chain complexes
• Grassmann algebra
• Laplacian and harmonic forms