= Degree of a holomorphic vector bundle
{title2=$\deg E=\deg\det E=\frac{i}{2\pi}\int_X\operatorname{tr}F_E$}
On a <compact Riemann surface>, the degree of a <holomorphic vector bundle> is the <degree of a line bundle> of its <determinant line bundle>. A <Hermitian metric on a holomorphic vector bundle> and its <Chern connection> express this degree by the trace-curvature integral. The value is independent of the metric because the normalized trace curvature represents the <First Chern class>.
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