Two vector bundles $E$ and $F$, are said two be S-equivalent if they have isomorphic gradients. My question is: Is it possible to caracterise this properity using transition functions?
Thanks



s-equivalence and transition function
Yes. Suppose for simplicity that $E$ and $F$ are extensions of two stable bundles with the same slope, say of ranks $r$ and $s$. Then you can choose their transition matrices of the form $\pmatrix{ (g\_{ij}) & (a\_{ij})\\ 0 & (h\_{ij})}$, with $(g\_{ij}) \in GL\_r(\mathcal{O})$ and $(h\_{ij}) \in GL\_s(\mathcal{O})$, and same for $F$. S-equivalence means that the cocycles $(g\_{ij})$ for $E$ and $F$ are cohomologous, and same for $(h\_{ij})$.


