Let \(\zeta\) be a \(t\)-dimensional real vector bundle over the real projective \(n\)-space \(P^n\). We say that \(\zeta\) is extendible (resp. stably extendible) to \(P^m\) for \(m \geq n\) if there is a vector bundle over \(P^m\) whose restriction to \(P^n\) is equivalent (resp. stably equivalent) to \(\zeta\), and denote by \(\text{span} \;\zeta\) the maximal number of linearly independent cross sections of \(\zeta\). In this paper three main theorems are proved (Theorems 1, 2 and 3).
In the first one the authors establish a relation between extendibility of \(\zeta\) which is stably equivalent to \(m\xi_n\) and \(\text{span} \;\ell\xi_n\), where \(\xi_n\) denotes the canonical line bundle over \(P^n\). For example, it is shown that if \(\zeta\) is stably equivalent to \((t+\ell)\xi_n\) for \(\ell \geq 1\), then \(\zeta\) is stably extendible to \(P^m\) for \(m \geq n\) if and only if \(\text{span}(a2^{\phi(n)}+t+\ell) \xi_m\geq a2^{\phi(n)}+\ell\) for some \(a\geq 0\). Here \(\phi(n)\) denotes the number of integers \(s\) such that \(0 < s \leq n\) and \(s \equiv 0, 1, 2\) or \(4 \;\text{mod} \;8\).
The second theorem deals with the existence of an immersion of \(P^n\) in \(\mathbb{R}^{n+k}\) and stable extendibilty of the normal bundle associated to such an immersion. The assertion is: Suppose that \(0 < k < 2^{\phi(n)}-n-1\) and \(\text{span}(n+k+1)\xi_m\geq n+1\) for \(m \geq n\). Then there is an immersion of \(P^n\) in \(\mathbb{R}^{n+k}\) and the normal bundle associated to this immesion is stably extendible to \(P^m\).
The last theorem gives a generalization of the result about \(\text{span} \;m\xi_n\) given by the second author in [J. Sci. Hiroshima Univ. Ser. A-I 32, 5–16 (1968; Zbl 0159.25103)]. Furthermore this paper contains some other interesting results.