  1-flat irreducible G-structures, equivalently, irreducible G-structures
admitting torsion-free affine connections, have been studied extensively in
differential geometry, especially in connection with the theory of affine
holonomy groups. We propose to study them in a setting in algebraic geometry,
where they arise from varieties of minimal rational tangents (VMRT) associated
to families of minimal rational curves on uniruled projective manifolds.
  We prove that such a structure is locally symmetric when the dimension of the
uniruled projective manifold is at least 5. By the classification result of
Merkulov and Schwachh\"ofer on irreducible affine holonomy, the problem is
reduced to the case when the VMRT at a general point of the uniruled projective
manifold is isomorphic to a subadjoint variety. In the latter situation, we
prove a stronger result that, without the assumption of 1-flatness, the
structure arising from VMRT is always locally flat. The proof employs the
method of Cartan connections. An interesting feature is that Cartan connections
are considered not for the G-structures themselves, but for certain geometric
structures on the spaces of minimal rational curves.
