  From the view of Heegaard splitting, it is known that if a closed orientable
3-manifold admits a distance at least three Heegaard splitting, then it is
hyperbolic. However, for a closed orientable 3-manifold admitting only distance
at most two Heegaard splittings, there are examples shows that it could be
reducible, Seifert, toroidal or hyperbolic. According to Thurston's
Geometrization conjecture, the most important piece of eight geometries is
hyperbolic. Thus to read out a hyperbolic 3-manifold from a distance two
Heegaard splittings is critical in studying Heegaard splittings.
  Inspired by the construction of hyperbolic 3-manifolds with a distance two
Heegaard splitting [Qiu, Zou and Guo, Pacific J. Math. 275 (2015), no. 1,
231-255], we introduce the definition of a locally large geodesic in curve
complex and furthermore the locally large distance two Heegaard splitting. Then
we prove that if a 3-manifold admits a locally large distance two Heegaard
splitting, then it is a hyperbolic manifold or an amalgamation of a hyperbolic
manifold and a seifert manifold along an incompressible torus, i.e., almost
hyperbolic, while the example in Section 3 shows that there is a non hyperbolic
3-manifold in this case. After examining those non hyperbolic cases, we give a
sufficient and necessary condition for a hyperbolic 3-manifold when it admits a
locally large distance two Heegaard splitting.
