  Chains of minimal degree rational curves have been used as an important tool
in the study of Fano manifolds. Their own geometric properties, however, have
not been studied much. The goal of the paper is to introduce an infinitesimal
method to study chains of minimal rational curves via varieties of minimal
rational tangents and their higher secants. For many examples of Fano manifolds
this method can be used to compute the minimal length of chains needed to join
two general points. One consequence of our computation is a bound on the
multiplicities of divisors at a general point of the moduli of stable bundles
of rank two on a curve.
