Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Abstract

Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

Crystal elasto-plasticity on the Poincaré half-plane

“…The Poincaré (upper complex) half-plane is the set H = {x + iy ∈ C, y > 0}, endowed with the metric (ds) 2 = (dx) 2 + (dy) 2 /y 2 , see Kilford (2008); Terras (2013). A complex representation of the cone of tensors C = C T > 0 is provided by associating to any C the complex number as in Folkins (1991), Parry (1998):…”

“…As in Folkins (1991), Parry (1998), Baggio et al (2019), here we use a natural parameterization of 2D strain space by means of the upper complex Poincaré half-plane H, one of the best known models of the hyperbolic plane (Kilford, 2008;Terras, 2013). On H the well-known Dedekind tessellation (Mumford et al, 2002;Ye et al, 2005) transparently displays the action of GL-symmetry on strain space, and thus on a crystal's energy landscape.…”

Volumes and distributions for random unimodular complex and quaternion lattices

“…The inequalities (3.21) are recognised as specifying the fundamental domain in the upper half plane model of hyperbolic geometry, up to details on the boundary; see e.g. [37]. Starting with r 1 = b 1 /b 0 , |r 1 | < 1, the recurrence (3.20) is to be iterated until |r j+1 | 1.…”

“…The factor dxdy/y 2 , in keeping with the remark below (3.21), is familiar as the invariant measure in the upper half plane model of hyperbolic geometry [37]. Distributions for the lengths of ||α|| and ||β|| can be computed by appropriate integrations over (3.24) and (3.25) [13].…”

Harmonic analysis on the proper velocity gyrogroup

“…It investigates and generalizes the notions of Fourier series and Fourier transforms. In the past two centuries, it has become a vast subject with applications in diverse areas as signal processing, quantum mechanics, and neuroscience (see [18] for an overview).…”