nLab
stable bundle

A vector bundle over a Riemann surface is stable (resp. semistable) if the ratio of its degree and its rank is bigger (resp. bigger or equal) than this ratio for any of its subbundles.

• wikipedia: stable vector bundle

Many references are at moduli space of bundles. The notion is introduced by David Mumford in the context of GIT in 1960-s:

• D. Mumford, J. Fogarty, F. Kirwan, (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, 3rd ed. Springer 1994
• G. Faltings, Stable G-bundles and projective connections, J. Alg. Geometry , 2 (1993) pp. 507–568 MR1211997 Zbl 0790.14019