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Pertinent Properties of Euclidean Space
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The most important topological spaces are manifolds. A topological space X is called an n-dimensional manifold if X is a Hausdorff space and each x∈X has a neighborhood N(x) that is homeomorphic to Euclidean n-space ℝn. Bernhard Riemann, a former student of Carl Gauss, was the first to coin the term ”manifold” in his extensive studies of surfaces in higher dimensions. In modern physics, our universe is a manifold. Einstein's general relative theory relies on a four-dimensional spacetime manifold. The string theory model of our universe is based on a ten-dimensional manifold R×M, where R denotes the four-dimensional spacetime manifold and M denotes the six-dimensional spacial Calabi-Yau manifold. Einstein's spacetime manifold is a generalized Riemannian manifold that uses the Ricci curvature tensor [32]. We do not expect the reader to be familiar with manifold theory and associated geometries. One reason for mentioning the subject of manifolds is that even Einstein's manifold, which tells us that that space and the gravitational field are one and the same, needs the locality of ℝ4 at each point of the manifold.
Implications of causality for quantum biology – I: topology change
Published in Molecular Physics, 2018
Let us briefly describe the topological properties that can be computed. A first step is to generate the underlying manifold. This step is provided by solving the quantum dynamical manifold equations (QDMEs) described below. These equations consist of 4D Minkowski spacetime generators for configuration coordinate fields (CCFs). The generators have a 1D time and 3D spatial structure. There can be multiple generators combining to generate a mass-spacetime manifold. The group of transformations of each generator is the Lorentz group and affine translations as well as conformal maps. The physical spaces of interest are the past and future causal cones. This (1 + 3)D decomposition focuses on 3D submanifolds evolving through time. A well-known example of the kind of results is that described for 2D manifolds spanned by triangles having faces, edges and vertices. If the 2D manifold has say k holes, thenThe 3D manifolds generated are considered to be spanned by a simplicial net consisting of tetrahedra. The number and type of holes in the generated manifolds can be computed using (co) homology [22]. Large biomolecules are often twisted and linked together. A quantity called the writhe can be defined reflecting the fact that a DNA molecule will coil if it is twisted. This leads to the formula [25]The linking number can be computed using a method going back to Gauss. These results can only be computed if the manifold can be computed. The spacetime current J can then be computed from the wave function thereby providing the basis for topology computations. Other topological characterisations of manifolds employ knot theory [26]. As a time-dependent problem, the manifold can evolve re-linking.