  In this {paper}, we consider holomorphic embeddings of $Gr(2,m)$ into
$Gr(2,n)$. We study such embeddings by finding all possible total Chern classes
of the pull-back of the universal bundles under these embeddings. Using the
relations between Chern classes of the universal bundles and Schubert cycles,
together with properties of holomorphic vector bundles of rank $2$ on complex
Grassmannians, we find conditions on $m$ and $n$ for which any holomorphic
embedding of $Gr(2,m)$ into $Gr(2,n)$ is linear.
