Uniformization of Kähler manifolds with vanishing Bochner tensor
\textit{S. Bochner} introduced in [Ann. Math., II. Ser. 50, 77-93 (1949; Zbl 0039.176)] a curvature tensor on a Hermitian manifold which corresponds to the Weyl conformal curvature tensor of a Riemannian manifold. A Riemannian manifold \(M^n\) whose Weyl conformal curvature tensor \((n > 3)\) or Schouten-Weyl tensor \((n = 3)\) vanishes, can be uniformized over \(S^n\) with respect to the group of conformal transformations \(\text{Conf} (S^n)\). The main result of the present paper states that a \(2n\)-dimensional Kähler manifold \(M\) whose Bochner tensor vanishes when \(n > 1\) or the tensor \(\widehat {R}\) (defined by covariant differentiation of the scalar curvature) vanishes when \(n = 1\), can be uniformized [see also \textit{R. S. Kulkarni}, J. Differ. Geom. 13, No. 1, 109-138 (1978; Zbl 0394.53018)] over \(Y^n_C\) with respect to \(G\), where \((G, Y^n_C)\) is one of the following: (1) a projective geometry \((\text{PU} (n + 1), \mathbb{C} P^n)\); (2) a similarity geometry \((\mathbb{C}^n \times (\text{U}(n) \times \mathbb{R}^+), \mathbb{C}^n)\); (3) a hyperbolic geometry \((\text{PU} (n,1), \mathbb{H}^n_C)\); (4) a projective-hyperbolic geometry \((\text{PU} (m,1) \times \text{PU} (n - m + 1), \mathbb{H}^m_C \times\mathbb{C} P^{n - m}),\) \(m = 1,2, \dots, n - 1\).
scientific article; zbMATH DE number 764290
scientific article