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Topological content of the Maxwell theorem on multipole representation of spherical functions. (English) Zbl 0914.58003

From the text: “A spherical function of degree \(n\) on the unit sphere in \(\mathbb{R}^3\) is the restriction to the sphere of a homogeneous harmonic polynomial of degree \(n\).
In the present paper the topological consequences of the following classical fact are discussed.
Theorem 1. The \(n\)th derivative of the function \(1/r\) along \(n\) constant (translation-invariant) vector fields in \(\mathbb{R}^3\) coincides on the sphere with a spherical function of degree \(n\). Any nonzero spherical function of degree \(n\) can be obtained by this constructoin from some \(n\)-tuple of nonzero vector fields. These \(n\) fields are uniquely defined by the function (up to multiplication by nonzero constants and permutation of the \(n\) fields).
The space of spherical functions of degree \(n\) is linear, of dimension \(2n+1\).
The set of functions representable by the multiple construction of the theorem is a priori highly nonlinear. The theorem implies that the image of the corresponding polylinear mapping is a linear space. The unicity statement can be reformulated in purely topological terms.
Theorem 2. The configuration space of \(n\) (virtually coinciding) indistinguishable points on the real projective plane (i.e., the \(n\)th symmetric power \(\text{Sym}^n(\mathbb{R} P^2)\)) is diffeomorphic to the real projective space of dimension \(2n\): \[ \text{Sym}^n(\mathbb{R} P^2)\approx \mathbb{R} P^{2n}. \] Remark. Theorem 2 is a relative of the projective Viète’s theorem \[ \text{Sym}^n(\mathbb{C} P^1)\approx \mathbb{C} P^n \] and is in a sense a quaternionic version of it.
Considering the Riemann sphere \(\mathbb{C} P^1\) as a two-fold covering of the real projective space, we construct as a corollary an algebraic mapping \(r: \mathbb{C} P^n\to \mathbb{R} P^{2n}\) of multiplicity \(2^n\) and generalize the classical theorem \(\mathbb{C} P^2/\text{conj}\approx S^4\) to higher dimensions”.

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
14P25 Topology of real algebraic varieties
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