Isomorphism of metrized line bundles are isomorphisms of line bundles which respect the metrics ; they are called isometries. Constructions from tensor algebra extend naturally to the framework of metrized line bundles, compatibly with isometries. The tensor product of two metrized line bundles $\overline{L}$ and $\overline{M}$ has a natural metrization such that $\left\|{s\otimes t}\right\|=\left\|{s}\right\|\left\|{t}\right\|$, if $s$ and $t$ are local sections of $L$ and $M$ respectively. Similarly, the dual of a metrized line bundle has a metrization, and the obvious isomorphism $L\otimes L^{\vee}\simeq\mathscr{O}_{X}$ is an isometry. Consequently, isomorphism classes of metrized line bundles on $X$ form an Abelian group $\overline{\operatorname{Pic}}(X)$. This group fits in an exact sequence

$$0\rightarrow\mathscr{C}(X)\rightarrow\overline{\operatorname{Pic}}(X)\rightarrow\operatorname{Pic}(X)\rightarrow 0,$$

where the first map associates to a real continuous function $h$ on $X$ the trivial line bundle endowed with the metric such that $\left\|{1}\right\|=e^{-h}$, and the second associates to a metrized line bundle the underlying line bundle. It is surjective when any line bundle has a metric (this certainly holds if $X$ has partitions of unity).

Similarly, we can consider pull-backs of metrized line bundle. Let $\varphi\colon Y\rightarrow X$ be a morphism of locally ringed spaces such that $\left|{\varphi^{*}f}\right|=\left|{f}\right|\circ\varphi$ for any $f\in\mathscr{O}_{X}$. Let $\overline{L}$ be a metrized line bundle on $X$. Then, there is a canonical metric on $\varphi^{*}L$ such that $\left\|{\varphi^{*}s}\right\|=\left\|{s}\right\|\circ\varphi$ for any section $s\in\Gamma(U,L)$. This induces a morphism of Abelian groups $\varphi^{*}\colon\overline{\operatorname{Pic}}(X)\rightarrow\overline{\operatorname{Pic}}(Y)$.