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A spherical projection of a complex Hilbert space is conformal iff it is the stereographic projection. (English) Zbl 1404.46019

The article deals with spherical projections (analogues and modifications of the Riemann and Poincaré stereographic projections) in a complex Hilbert spaces. More precisely, with maps (nonlinear projections) \(H \to SP(\theta_s,R)\), where \(H\) is a complex Hilbert space, \(SP(\theta_su,R)\) a sphere with the center at a point \(O = (\theta_su,0)\) and radius \(R\), in a new complex Hilbert space \(H_N = \{(cu,z):\;c \in {\mathbb C}, z \in H\}\) (\(u\) is a fixed vector from \(H\), \(\langle u,u \rangle = 1\)) with the usual algebraic operations and the scalar product \(\langle (c_1u,z_1),(c_2u,z_2) \rangle = \langle z_1,z_2 \rangle + c_1 \overline{c}_2\). It is assumed that a “projection point” \(P = (\gamma_su,0)\) (\(\gamma_s \in {\mathbb C} \setminus \{0\}\)) and the point \(O\) satisfy some natural conditions. Under these conditions, the authors prove the following.
Theorem. In a complex Hilbert space, the following conditions are equivalent: (i) A spherical projection is the stereographic projection. (ii) The spherical projection is conformal. (iii) Any two triangles \(PQ\widehat{Q}\) and \(P\widehat{Z}Z\) are similar.
The authors state that various so-called “compactifications” found in the literature are generated with the help of special members of these nonlinear projections. At the end of the article, this idea is discussed on several examples illustrating the results provided.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
51N15 Projective analytic geometry
14M27 Compactifications; symmetric and spherical varieties
14N05 Projective techniques in algebraic geometry
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