  For a fixed positive integer $g$, we let ${\mathcal P}_g = \big\{Y\in
{\mathbb R}^{(g,g)} | Y= {}^tY>0 \big\}$ be the open convex cone in the
Euclidean space ${\mathbb R}^{g(g+1)/2}$. Then the general linear group
$GL(g,{\mathbb R})$ acts naturally on ${\mathcal P}_g$ by $A\star Y= AY {}^tA$
($A\in GL(g,{\mathbb R}), Y\in {\mathcal P}_g$). We introduce a notion of
polarized real tori. We show that the open cone ${\mathcal P}_g$ parametrizes
principally polarized real tori of dimension $g$ and that the Minkowski domain
${\mathfrak R}_g= GL(g,{\mathbb Z})\backslash {\mathcal P}_g$ may be regarded
as a moduli space of principally polarized real tori of dimension $g$. We also
study smooth line bundles on a polarized real torus by relating them to
holomorphic line bundles on its associated polarized real abelian variety.
