to be merged with stable vector bundle.
(see also Chern-Weil theory, parameterized homotopy theory)
A vector bundle over a Riemann surface is called 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.
This means that it is a stable point under the action of automorphisms.
Many references are at moduli space of bundles. The notion was 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
Gerd Faltings, Stable G-bundles and projective connections, J. Alg. Geometry , 2 (1993) pp. 507–568 MR1211997 Zbl 0790.14019
See also