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Riemann Symbols (Curvature Tensors)
Published in Bhaben Chandra Kalita, Tensor Calculus and Applications, 2019
The importance of intrinsic multidimensional differential geometry has found its place in the study of general theory of relativity. The general theory of relativity is inherently related to the geometry of curved space due to the effects of gravity. For this reason, Einstein needs to deal with four-dimensional space–time continuum in support of “absolute differential calculus” or tensor calculus. The Riemannian geometry associated with covariant differentiation through the fundamental tensor gij characterizing the space is properly suited for the curved space of general theory of relativity. This demands, though not general, an important notion (or concept), namely, Riemannian symbols or curvature tensors.
Celestial Mechanics and Astrodynamics
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
Einstein’s general theory of relativity speculated that the space-time around a massive body is distorted and curved, and such curved space-time can be described mathematically by Riemann geometry, as illustrated in Figure 12.18. In 1915, Einstein published his celebrated paper on the advance of perihelion precession of Mercury using general relativity based on the approximate gravitational field around a spherically symmetric, non-rotating, non-charged mass. On December 22, 1915, while still serving a war stationed on the Russia front of World War I, Karl Schwarzschild sent a letter to Einstein and presented his exact solution of Einstein’s field equation in polar form. In 1916, Einstein replied:
Les vertus des défauts: The scientific works of the late Mr Maurice Kleman analysed, discussed and placed in historical context, with particular stress on dislocation, disclination and other manner of local material disbehaviour
Published in Liquid Crystals Reviews, 2022
Figure 20 shows pentagons failing to tile a two dimensional space. The idea extends to three dimensions, with pentagons replaced by icosahedra. However, imagining it without the aid of group theory is almost impossible. The curved spaces may have positive curvature and thus be elliptic spaces, or negative curvature (hyperbolic spaces). In the curved space, there are crystallographic groups analogous to those of Euclidean space, and the difficulty involves the imperfect mapping between the space groups in the curved and flat spaces. In [172], inter alia Maurice wrestles with some apparently universal features of glassy materials, including the linear specific heat at low temperatures, which should be related to the existence of low lying acoustic modes which he suggested were analogous to the rotons found in liquid Helium.
Industrial statistics and manifold data
Published in Quality Engineering, 2020
Enrique del Castillo, Xueqi Zhao
Quality Engineers working in industry must be acquainted with new ways to analyze not only larger datasets, but frequently, more complex datasets. In this article we address the case the complexity of a dataset occurs because the data follow, apart of measurement error, a lower dimensional manifold, which can be understood informally as a curved space which, when looked over a small domain around any point in the space, resembles Euclidean space (as a typical example, consider the surface of the Earth or a spheroid in ). We will focus on two of the main subfields of Industrial Statistics: Statistical Process Control (i.e., monitoring, hereafter referred to as SPC) and Experimental Design and Analysis. We are aware that the subject of manifold data in industrial statistics is a wider theme than these subtopics and we do not aim at a comprehensive review; the subfields we concentrate are admittedly limited to our personal research interests. In the machine learning literature, manifold learning is a topic that has received considerable attention, and similarly, “shape analysis” and “statistical shape analysis (SSA)” of 3D objects, whose surface is a 2-dimensional manifold, have been intensively studied in the computer vision and statistics fields, respectively, over the last two decades. We will discuss intersections between Industrial Statistics and these fields when appropriate. We concentrate in manufacturing industry, where discrete parts are produced and measured.