  A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive
if the real part of the holomorphic volume form restricted to it is positive.
An exact isotopy class of positive Lagrangian submanifolds admits a natural
Riemannian metric. We compute the Riemann curvature of this metric and show all
sectional curvatures are non-positive. The motivation for our calculation comes
from mirror symmetry. Roughly speaking, an exact isotopy class of positive
Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics
on a holomorphic vector bundle. The latter space is an infinite-dimensional
analog of the non-compact symmetric space dual to the unitary group, and thus
has non-positive curvature.
