  We give a real-analytic section for the Teichm\"uller projection onto the
VMO-Teichm\"uller space by using the variant of Beurling-Ahlfors extension by
heat kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this
result, we prove that the VMO-Teichm\"uller space can be endowed with a real
Banach manifold structure that is real-analytically equivalent to its complex
Banach manifold structure. We also obtain that the VMO-Teichm\"uller space
admits a real-analytic contraction mapping.
