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Spectral geometry
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
Let s be a spin structure and let σ be a G structure on M. Assume M admits a metric of positive scalar curvature g. The refined bordism groups MSpinm+(BG) are defined by introducing the equivalence relation (M,g,s,σ) ⋍ 0 if there exists a compact Riemannian manifold N with boundary M such that σ and s extend over N and so that the metric gextends over N as a metric gN of positive scalar curvature which is product near the boundary. We say that metrics of positive scalar curvature g0 and g1 on a manifold M are bordant if () [(M,g0,s,σ)]=[(M,g1,s,σ)].
Orthogonal Expansions in Curvilinear Coordinates
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
respectively. The scalar curvature for ℝN is zero. This is independent of what curvilinear coordinate system is used. For a sphere in any dimension, the scalar curvature is constant. In the case when M = 2 and N = 3, the scalar curvature is twice the Gaussian curvature.
V
Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[astrophysics/astronomy, computational, optics] Description of gravitational interaction, including gravitational waves, and specifically the interaction with electromagnetic radiation known as Einstein’s field equations under conditions at absolute zero temperature, expressed as Ric − (1/2) gµρR + RµρΛ = (8π/c4)Gτµρ, where Ric=Rμρ=Rμνρν the Ricci curvature tensor; a contracted Riemann tensor (Rμνρν), which vanishes in vacuum, τµρ the stress energy tensor, gμρ=(∂/∂yμ)(∂/∂yρ)((1/2)F2) the metric tensor derived from a matrix diagonal in Finsler structure (the metric tensor is directly linked to the Ricci factor, however with only half the number of independent components: 10) and F the Finsler structure; with property: F(x, λy) = λF(x, y), operating in Finsler geometry, not Riemann geometry, subsequently: Λ the cosmological constant (the energy density of space vacuum, assuming a stationary universe), R the scalar curvature, G the gravitational constant, and c the speed of light. Einstein had to admit to an error in his assumption of the universe, being an expanding universe, as revealed by Edwin Hubble (1889–1953) in 1915; leading to a significant adjustment for the cosmological constant (Λ). Additionally, under the condition of absolute zero the energy density tensor (τµρ) drops to zero as well. Defined by Albert Einstein (1879–1955), as part of his general theory ofrelativity, in 1915.
Spacelike submanifolds of codimension two in anti-de Sitter space
Published in Applicable Analysis, 2019
It is well known that there are three kinds of Lorentzian space forms. The Lorentz–Minkowski space is a flat Lorentzian space form and the de Sitter space is a Lorentzian space form with a positive curvature. Several articles are devoted to the study of submanifolds in these two Lorentzian space forms [1–3]. A Lorentzian space form with a negative curvature is called an anti-de Sitter space which has rich mathematical and physical properties. This space is a maximally symmetric semi-Riemannian manifold with constant negative scalar curvature, and it is also one of the vacuum solutions of Einstein’s field equation in the theory of relativity. There is a conjecture in physics that the classical gravitation theory on anti-de Sitter space is equivalent to the conformal field theory on the ideal boundary of anti-de Sitter space. It is called the AdS/CFT correspondence or the holographic principle by E.Witten. In mathematics, this conjecture is that the extrinsic geometric properties on submanifolds in anti-de Sitter space have corresponding Gauge theoretic geometric properties in its ideal boundary. Therefore, it is necessary to investigate the submanifolds in anti-de Sitter space.