Almost complex and complex structures on real projective Stiefel manifolds
The projective Stiefel manifold \(X_{n,r}\) is the quotient of the Stiefel manifold \(V_{n,r}\) of orthonormal \(r\)-frames in \(\mathbb{R}^n\) by the \(\mathbb{Z}/2\) action induced from the antipodal action on \(\mathbb{R}^n\). The main result of the paper is the following Theorem. The manifold \(X_{n,r}\) admits an almost complex structure (compatible with the usual differentiable structure) if and only if \(n\) is arbitrary and \(r\equiv 0\pmod 4\) or \(n\) is even and \(r\equiv 3\pmod 4\). It is shown that in fact the manifolds \(X_{n,r}\) of the first type \((r\equiv 0\pmod 4)\) admit complex structures. This is unknown for general manifolds \(X_{n,r}\) of the second type \((r\equiv 3\pmod 4)\).
scientific article; zbMATH DE number 1279146
scientific article