Nilpotent structures of neutral 4-manifolds and light-like surfaces
Let \(M\) be an oriented 4-dimensional manifold with neutral metric \(h\). Let \(U_0(\Lambda_{\pm}^2 TM)\) be bundles, contained in subbundles \(\Lambda_{\pm}^2 TM\) of rank 3 in the 2-fold power \(\Lambda^2 TM\) of \(TM\), whose fibers are light-like cones. A \((1,1)\)-tensor field \(N\) of \(M\) is an almost nilpotent structure of \(M\) if \(N\) gives a nilpotent structure of the tangent space of \(M\) at each point. The author studies almost nilpotent structures of \((M,h)\) and light-like surfaces in \((M,h)\). An almost nilpotent structure \(N\) of \(M\) gives a light-like 2-plane of the tangent space at each point of \(M\) which can be considered as a light-like two-dimensional distribution \(\mathfrak{D}\). It is shown there exists the interdependence between \(N\) and the corresponding section \(\Theta\) of either \(U_0(\Lambda_{+}^2 TM)\) or \(U_0(\Lambda_{-}^2 TM)\). Especially, the author discusses the distributions \(\mathfrak{D}\) on conditions that they are involutive. This discussion makes possibly also to study light-like surfaces in \(M\) which are integral surfaces of involutive distributions \(\mathfrak{D}\). These light-like surfaces with local horizontal lifts are characterized using the curvature tensors. It is shown that they are analogues of isotrophic minimal surfaces in Riemannian 4-manifolds. An analogy between nilpotent structures of neutral 4-manifolds from one side, and complex and para-complex structures from other side are established. For the entire collection see [Zbl 1497.53003].
scientific article; zbMATH DE number 7632937
scientific article