# Lectures on vector bundles over Riemann surfaces by Robert C. Gunning

0 is defined by a non-singular holomorphic matrix- valued function over V , that is, by an element © e GL(s1 + s2, &V) . By Lemma 7, the matrix © can be approxi- mated uniformly over the compact subset U C V by a non-singular holomorphic matrix-valued function over V1 U V2 ; thus there is an element ©0 E GL(s1 + s2, S p c U the matrix such that for all points ) 2 1 ©(p)©0(p)-1 c A , where A is an open neighbor- hood of the identity V U V I c GL(s1 + s2, c) In particular, select A .

Coordinate, for example, D C Ui (z. E cffzjf < 2) 0 e H-(1P, /S X (m, B")) complex vector bundle ment , and let 7P open subsets homeomorphic to the disc be the standard com- In terms of the zi is a holomorphic, non-singular 012(z1) matrix-valued function in the annulus < 1,11 < 2 ; and the D: 2 Laurent expansions of the various entries yield a matrix a of rational functions on which approximate 7P any compact subset of the annulus 012 uniformly on (Recall that a rational D . ) 1P , with singularities at z2 = 0 , such that < I zi and < e is non- and 12(p)©(p)-1 c t, 3 2 p e D , where D is an open neighborhood of the I e GL(m,C) Let D C D be the open subset homeomorphic to the disc IV IV ` .

Cm X m and an I e GL(m,C) ; let (For the definitions and ele- mentary properties of the matrix exponential function, see for instance C. Chevalley, Theory of Lie Groups, I, (Princeton Univ. ) neighborhoods of the origin in Cm X m exp Xl . exp X2 e AO whenever Xi a Di Let Dl, D2 C DC be open such that . it 0'm X m) S2j C r0(Uj, be the open subset of that Banach space defined by S2j = (G e rG(Uj, (¢ m X m) I G(p) a Dj for all p e Uj ) . It is then possible to define a mapping LILm xm) 121 ® 122 -> rG(U, `Y: by putting T(G1,G2)=exp-'(expG1- expG2); and it is evident that T is a continuous mapping from the open subset 12l ® 122 C r0(U1, 0) ®rG(U2, C9 ) into the Banach space r0(U, (QmXm) .