Poincaré automorphisms for nondegenerate CR quadrics
Let \(\langle , \rangle\) be (a nondegenerate vector valued Levi form, i.e.) an \(\mathbb{R}^k\) valued Hermitian form in \(\mathbb{C}^n\) satisfying (i) if \(\langle z, b \rangle = 0\) for all \(z\) in \(\mathbb{C}^n\), then \(b = 0\); (ii) if \(f (\langle z,z \rangle) \equiv 0\) for some linear functional \(f\) in \((\mathbb{R}^k)'\), then \(f = 0\). The set \(Q = \{(z,w)\) in \(\mathbb{C}^n \times \mathbb{C}^k \mid \text{Im} (w) = \langle z,z \rangle\}\) is said to be a nondegenerate quadric in \(\mathbb{C}^{n + k}\). The authors study some automorphisms of \(Q\) which are called Poincaré automorphisms, and furnish some explicit formulas for them. Moreover they show that such automorphisms generalize linear fractional transformations. The last part of the paper is devoted to the discussion of some questions concerning automorphisms of quadrics.
scientific article; zbMATH DE number 537308