Minimal surfaces of pseudo-Euclidean spaces with geodesic normal sections
Let \(M\) be a pseudo-Riemannian surface of index \(r\) which lies in an \(m\)-dimensional pseudo-Euclidean space \(E^m_s\) of index \(s\). We define a normal section of \(M\) at a point \(p\in M\) in a direction \(t\) \((\neq 0)\in T_p M\), the tangent space of \(M\) at \(p\), in the same way as in the Riemannian case: Let \(E(p; t)\) be the affine space generated by \(t\) and the normal space \(T^\perp_p M\) of \(M\) at \(p\). The intersection of \(M\) and \(E(p; t)\) gives rise to a curve in a neighbourhood of \(p\) which is called the normal section of \(M\) at \(p\) in the direction \(t\). An isometric immersion \(i: M\to E^m_s\) is said to be planar geodesic if the image of each geodesic of \(M\) lies in a 2-plane of \(E^m_s\). In this paper, minimal surfaces of pseudo-Euclidean space with geodesic normal sections are studied. We prove that complete connected minimal surfaces in a 5-dimensional pseudo-Euclidean space with geodesic normal sections are totally geodesic or flat quadric.
scientific article; zbMATH DE number 897456