Complex manifolds modeled on a complex Minkowski space
The author studies some problems of the differential geometry of complex Finsler manifolds. He defines a complex Finsler metric on \(\mathbb{C}^n\) as a function \(f : \mathbb{C}^n \to R\) having the properties: i) \(f(\xi) \geq 0\), ii) \(f(\xi)\) is \(C^\infty\) on \(\mathbb{C}^n \setminus \{0\}\) and continuous on \(\mathbb{C}^n\), iii) \(f(\lambda \xi) = |\lambda|^2 f(\xi)\), \(\lambda \in \mathbb{C}\), and iv) the Hermitian matrix \(|{\partial^2 f/\partial \xi^\alpha\partial \overline {\xi}^\beta} |\) is positive definite. This notion is extended to complex manifolds \(M\) using the tangent bundle TM. Thus, a complex Finsler manifold is a pair \(^\mathbb{C} F^n = (M,F)\), where \(M\) is a complex manifold and \(F\) is a complex Finsler metric on \(M\). For \(^\mathbb{C} F^n\) the notion of Finsler connection \(\nabla\), the Kähler condition for \(\nabla\) and complex manifolds modeled on a complex Minkowski space are studied.
scientific article; zbMATH DE number 784737
scientific article