Hurwitz theorem and parallelizable spheres from tensor analysis

By using tensor analysis, we find a connection between normed algebras and the parallelizability of the spheres S-1, S-3 and S-7. In this process, we discovered the analog of Hurwitz theorem for curved spaces and a geometrica l unified formalism for the metric and the torsion. In order to achieve the se goals we first develop a proof of Hurwitz theorem based on tensor analys is. It turns out that in contrast to the doubling procedure and Clifford al gebra mechanism, our proof is entirely based on tensor algebra applied to t he normed algebra condition. From the tersor analysis point of view our pro of is straightforward and short. We also discuss a possible connection betw een our formalism and the Cayley-Dickson algebras and Hopf maps.