Quotient Hermitian metric ($h_Q([v],[w])=h_E((I-{\pi }_S)v,(I-{\pi }_S)w)$)

For a smooth subbundle $S$ of a Hermitian vector bundle $E$, orthogonal projection identifies $E/S$ smoothly with $S^{\perp }$. The quotient metric is $h_Q([v],[w])=h_E((I-{\pi }_S)v,(I-{\pi }_S)w)$. It is independent of representatives and positive definite. For a holomorphic subbundle the resulting smooth quotient metric is defined on the holomorphic quotient, but the orthogonal identification itself need not be holomorphic.