Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher’s book “Vector Bundles and K theory”, the construction of sphere bundle $S(E)$ can be done without any inner product on fibers. It is a result of the equivalent relation on each punctured fiber: $vsim w$ if $v=lambda w$ for some positive scalar $lambda$.
We define an equivalent relation on the set of all vector bundles over $X$ as follows: $E$ is equivalent to $F$ if their corresponfing sphere bundles $S(E)$ and $S(F)$ are isomorphic fiber bundles.
Is the above equivalent relation compatible to direct sum and tensor product of vector bundles in the following sense;
If $E_1,E_2$ are equivalent bundles and $F_1,F_2$ are also equivalent bundles in the above sense, are $E_1oplus F_1, E_2 oplus F_2$ are equivalent bundles? What about the tensor product, instead of direct sum?
Can this equivalent relation define a kind of $K$ theory?