Summary: A complete decomposition of the space \({\mathcal R}(V^{p\otimes q})\) of curvature tensors over tensor products of vector spaces into simple modules under the action of the group \(G= \text{GL}(p,\mathbb R)\otimes\text{GL}(q,\mathbb R)\) is given. We use these results to study the geometry of manifolds with Grassmann structure and Grassmann manifolds endowed with a connection whose torsion is not zero. We show that \(\text{Osc}^rM\) is an example of a manifold with Grassmann structure. Owing to this fact, we consider results of Miron, Atanasiu, Anastasiei, Čomić and others from a representation theoretic point of view and connect them with some results of Alekseevsky, Cortes, and Devchand, as well as of Machida and Sato, and others. New examples of connections with torsion defined on four-dimensional Grassmann manifolds are given. Symmetries of curvatures for half-flat connections are also investigated. We use algebraic results to reveal obstructions to the existence of corresponding connections.