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Generalized plane wave manifolds. (English) Zbl 1143.53337

\(M=(\mathbb{R}^m,g)\) is a generalized plane wave manifold if \( \nabla_{\partial x_i}\partial x_j=\sum_{k>\max(i,j)} \Gamma_{ij}^k(x_1,\dots,x_{k-1})\partial x_k\), where \(x_i\) are usual coordinates on \(\mathbb{R}^m\). The authors investigate the geometry of generalized plane wave manifolds by means of natural curvature operators (such as Jacobi, Szabó, skew-symmetric curvature operator) point of view. They show that generalized plane wave manifolds are complete, strongly geodesically convex, nilpotent Osserman, nilpotent Szabó, nilpotent Ivanov-Petrova, Ricci flat and Einstein. They also show that such manifolds have nilpotent holonomy groups and that all the local Weyl scalar invariants of these manifolds vanish. The authors construct isometry invariants on certain families of these manifolds which are not of Weyl type. This family of manifolds contains examples of manifolds which are \(k\)-curvature homogeneous but not locally homogeneous and which are weakly \(1\)-curvature homogeneous but not \(1\)-curvature homogeneous. The paper provides by other interesting examples in this family of manifolds which show the difference between notions of time-like and space-like Jordan Osserman, time-like and spacelike Jordan Szabó and timelike and space-like Jordan Ivanov-Petrova manifolds.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B20 Local Riemannian geometry
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