In 1962, Adams proved that there do not exist $\rho(n)$ linearly independent vector fields on the sphere $S^{n-1}$, where $\rho(n)$ is the Hurwitz-Radon number. I wonder if this is still true in the case of quasi-sphere (a sphere in the pseudo-Euclidean space $\mathbb{R}^n_{\nu}$ of index $\nu$).
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2$\begingroup$ @ArunDebray these questions are not smooth deformation retract invariants. $\mathbf{R}^3-\{0\}$ is parallelizable but is SDR to the 2-sphere... $\endgroup$– YCorFeb 24, 2018 at 17:08
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4$\begingroup$ @YCor Are not they just diffeomorphic to some $\mathbb R^i\times S^j$? $\endgroup$– მამუკა ჯიბლაძეFeb 24, 2018 at 17:39
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$\begingroup$ @მამუკაჯიბლაძე why do you ask me? rather ask the OP. I mentioned that there are trivial counterexamples, it's enough for me at the moment. $\endgroup$– YCorFeb 24, 2018 at 18:15
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$\begingroup$ Thanks to @მამუკაჯიბლაძე. It is a better question, is there a similar bound for vector fields on $\mathbb{R}^{\nu}\times S^{m-1}$. $\endgroup$– ArimakatFeb 25, 2018 at 16:47
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