  Topological quantum liquids contain internal degrees of freedom that are
coupled to geometric response. Yet, an explicit and microscopic identification
of geometric response remains difficult. Here, taking notable fractional
quantum Hall (FQH) states as typical examples, we systematically investigate a
promising protocol -- the Dehn twist deformation on the torus geometry, to
probe the geometric response of correlated topological states and establish the
relation between such response and the universal properties of pertinent
states. Based on analytical derivations and numerical simulations, we find that
the geometry-induced Berry phase encodes novel features for a broad class of
FQH states at the Laughlin, hierarchy, Halperin and non-Abelian Moore-Read
fillings. Our findings conclusively demonstrate that the adiabatic Dehn twist
deformation can faithfully capture the geometry of elementary FQH droplets and
intrinsic modular information including topological spin and chiral central
charge. Our approach provides a powerful way to reveal topological orders of
generic FQH states and allows us to address previously open questions.
