Reduction of \(\beta\)-integrable 2-Segre structures
Let \(M\) be a smooth manifold of dimension \(mn\) with an \((m,n)\)-Segre structure. By definition, an \((m,n)\)-Segre structure on \(M\) is a smoothly varying family \(S\) of cones \(S_p \subset T_pM\) in the tangent spaces \(T_pM\) of \(M\), each of which is linearly isomorphic to the \((m,n)\)-Segre cone of simple vectors in the tensor product \(\mathbb R^m \otimes\mathbb R^n\). A linear subspace in a tangent space \(T_pM\) of \(M\) is called a \(\beta\)-plane if it is isomorphic to a linear subspace in \(\mathbb R^m \otimes\mathbb R^n\) of the form \(x \otimes\mathbb R^n\) for some \(x \in\mathbb R^m\). An immersed submanifold \(\Sigma \subset M\) whose tangent planes are all \(\beta\)-planes and which is maximal in the sense of inclusion is called a \(\beta\)-surface. A Segre structure is called \(\beta\)-integrable, if every \(\beta\)-plane is tangent to a unique \(\beta\)-surface. NEWLINENEWLINENEWLINEThe author studies the problem of reducing the \(G\)-structure associated to certain types of Segre structures to torsion-free structures. In his main theorem he proves that locally every \(\beta\)-integrable \((2,n)\)-Segre structure \(S\) can be reduced to a torsion-free \(S^1 \cdot \text{GL}(n,\mathbb R)\)-structure.
scientific article; zbMATH DE number 6192447