On the classification of almost product structures on a manifold with linear connection (Q1588976)

There is a natural bijection between the \(\text{GL} (n, \mathbb{R}) \otimes \text{GL}(m - n, \mathbb{R})\)-structures on an \(m\)-differentiable manifold \(M\) and the pairs \(({\mathcal H}, {\mathcal V})\) of complementary horizontal and vertical distributions of dimension \(n\) and \(m - n\) on \(M\). In [\textit{A. P. Norden}, The theory of compositions, Itogi Nauki Tekh., Probl. Geom. 10, 117-147 (1978; Zbl 0418.53009)] was given a partial classification of such structures using the concept of parallel displacement of \(\mathcal H\) and \(\mathcal V\). In the present paper the author continues these studies, reducing the problem of classification to a decomposition of tensor product representations into irreducible components. In this way he finds new classes of \(\text{GL} (n, \mathbb{R}) \otimes \text{GL}(m - n, \mathbb{R})\)-structures, giving also a geometrical description of them.