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Study of Some Fundamental Projective Quadrics Associated with a Standard Pseudo-Hermitian Space H p,q

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This self-contained short note deals with the study of the properties of some real projective compact quadrics associated with a a standard pseudo-hermitian space H p,q , namely \({\widetilde{Q(p, q)}, \widetilde{Q_{2p+1,1}}, \widetilde{Q_{1,2q+1}}, \widetilde{H_{p,q}}. \, \widetilde{Q(p, q)}}\) is the (2n – 2) real projective quadric diffeomorphic to (S 2p–1 × S 2q–1)/Z 2. inside the real projective space P(E 1), where E 1 is the real 2n-dimensional space subordinate to H p,q . The properties of \({\widetilde{Q(p, q)}}\) are investigated. \({\widetilde{H_p,q}}\) is the real (2n – 3)-dimensional compact manifold-(projective quadric)- associated with H p,q , inside the complex projective space P(H p,q ), diffeomorphic to (S 2p–1 × S 2q–1)/S 1. The properties of \({\widetilde{H_{p,q}}}\) are studied. \({\widetilde{Q_{2p+1,1}}}\) is a 2p-dimensional standard real projective quadric, and \({\widetilde{Q_{1,2q+1}}}\) is another standard 2q-dimensional projective quadric. \({\widetilde{Q_{2p+1,1}} \cup \widetilde{Q_{1,2q+1}}}\), union of two compact quadrics plays a part in the understanding of the "special pseudo-unitary conformal compactification" of H p,q . It is shown how a distribution yD y , where \({y \in H\backslash\{0\},H}\) being the isotropic cone of H p,q allows to \({\widetilde{H_{p+1,q+1}}}\) to be considered as a "special pseudo-unitary conformal compactified" of H p,q × R. The following results precise the presentation given in [1,c].

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Correspondence to Pierre Anglès.

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This paper is dedicated to Jaime Keller for his birthday, [November 10th, 2009].

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Anglès, P. Study of Some Fundamental Projective Quadrics Associated with a Standard Pseudo-Hermitian Space H p,q . Adv. Appl. Clifford Algebras 21, 233–246 (2011). https://doi.org/10.1007/s00006-010-0257-1

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