On the linearly independent vector fields on Grassmann manifolds (Q2252604)

Let \(n=2^{4\alpha +\beta}\cdot (2s+1)\), where \(\alpha,s\in{\mathbb{N}}\) and \(\beta\in\{ 0,1,2,3\}\). Consider the sphere \(S^{n-1}\). It is well known that this sphere admits \(\theta(n)=2^{\beta}+8\alpha-1\) linearly independent vector fields. In an old result by Adams it was proved that this number cannot be improved. The existence of linearly independent vector fields on the sphere is related to Clifford modules and skewsymmetric anticommuting complex structures on \({\mathbb{R}}^n\). In the present paper, the Grassmann manifolds \(G_k(V)\) for even \(n={\mathrm{dim}}V\) and odd \(k\in{\mathbb{N}}\) are studied. It is proved that they also admit \(\theta(n)\) linearly independent tangent vector fields. It is an open problem whether this result can be improved.