  In flat spacetime, the vacuum neutrino flavour oscillations are known to be
sensitive only to the difference between the squared masses, and not to the
individual masses, of neutrinos. In this work, we show that the lensing of
neutrinos induced by a gravitational source substantially modifies this
standard picture and it gives rise to a novel contribution through which the
oscillation probabilities also depend on the individual neutrino masses. A
gravitating mass located between a source and a detector deflects the neutrinos
in their journey, and at a detection point, neutrinos arriving through
different paths can lead to the phenomenon of interference. The flavour
transition probabilities computed in the presence of such interference depend
on the individual masses of neutrinos whenever there is a non-zero path
difference between the interfering neutrinos. We demonstrate this explicitly by
considering an example of weak lensing induced by a Schwarzschild mass. Through
the simplest two flavour case, we show that the oscillation probability in the
presence of lensing is sensitive to the sign of $\Delta m^2 = m_2^2 -m_1^2$,
for non-maximal mixing between two neutrinos, unlike in the case of standard
vacuum oscillation in flat spacetime. Further, the probability itself
oscillates with respect to the path difference and the frequency of such
oscillations depends on the absolute mass scale $m_1$ or $m_2$. We also give
results for realistic three flavour case and discuss various implications of
gravitationally modified neutrino oscillations and means of observing them.
