Regular maps on Cartesian products and disjoint unions of manifolds (Q1688201)

For a smooth manifold \(M\) and \(k\geq2\), a map \(f:M\rightarrow\mathbb F^N\) (where either \(\mathbb F=\mathbb R\) or \(\mathbb F=\mathbb C\)) is called \(k\)-regular if \(f(x_1),f(x_2),\ldots,f(x_k)\) are linearly independent vectors for arbitrary pairwise different points \(x_1,x_2,\ldots,x_k\in M\). For a given manifold \(M\) and a given integer \(k\geq2\), a natural question arises: what is the smallest \(N\) such that there is a \(k\)-regular map \(f:M\rightarrow\mathbb F^N\)? This problem was widely considered in the past, and the most studied cases were the ones when \(M\) is either some Euclidean space or a sphere. In the present paper the author considers \(2\)-regular maps \(f:M\rightarrow\mathbb F^N\) and establishes some lower bounds for \(N\) in the cases when \(M\) is a sphere, a (real, complex or quaternionic) projective space, and a product of such spaces. He also generalizes a previous theorem concerning the case of Euclidean spaces. Some of the obtained results are the following: {\parindent=0.7cm \begin{itemize}\item[--] if there is a \(2\)-regular map \(f:\mathbb R\mathrm P^m\rightarrow\mathbb R^N\), and \(2^i\leq m<2^{i+1}\), then \(N\geq2^{i+1}+1\); \item[--] if there is a \(2\)-regular map \(f:\mathbb C\mathrm P^m\rightarrow\mathbb R^N\), and \(2^i\leq m<2^{i+1}\), then \(N\geq2^{i+2}\); \item[--] if there is a \(2\)-regular map \(f:\mathbb C\mathrm P^m\rightarrow\mathbb C^N\), and \(m\geq4\), then \(N\geq2m\). \end{itemize}} In addition to this, the author introduces a generalization of \(k\)-regular maps to disjoint unions of manifolds, and obtains some results in this regard.