  The $L^1$-$L^\infty$ geometry is the Finsler geometry of the Teichm\"uller
space by the Teichm\"uller metric and the $L^1$-norm function of holomorphic
quadratic differentials. In this paper, aiming to develop the
$L^1$-$L^\infty$-geometry and the differential geometry on the Teichm\"uller
space, we formulate the second order infinitesimal structures (the
infinitesimal structures on the (co)tangent bundles) over the Teichm\"uller
space. We will give model spaces of the second order infinitesimal spaces.
  By applying our formulation, we give affirmative answers to two folklore. We
first show that the map from the space of holomorphic quadratic differentials
to the tangent bundle defined by Teichm\"uller Beltrami differentials is a
real-analytic diffeomorphism on every stratum in the space of holomorphic
quadratic differentials. Second, we show that the Teichm\"uller metric is
real-analytic on the image of each stratum. We also observe a new duality
between the Teichm\"uller metric and the $L^1$-norm function at the
infinitesimal level.
