I will illustrate with an example of the quantum hall effect. The quantum hall effect, of course, demonstrates the quantization of resistance at low temperatures in a 2-dimensional substrate that is pierced by a perpendicular magnetic field. Why is this? Well, without going into too much detail, there are essentially two parameters you can vary in a quantum hall system. So, your quantum hall system has a two-parameter family of Hamiltonians. Both of these parameters happen to be periodic, i.e. if you slowly increase one parameter, you will eventually obtain the same system. Thus, your family of Hamiltonians forms a parameter space that is actually a torus.
Now, to each point of your torus you glue the associated eigenspace. To simplify matters, we set temperature to zero so that all of the occupied eigenstates lie below a certain energy, the Fermi energy, which allows us to consider the eigenspace to be finite-dimensional by just deleting all of the unoccupied eigenstates. We need some other 'niceness' requirements as well to ensure that the dimension of the eigenspace is constant over the parameter space, that there is no degeneracy of eigenvalues, etc.
Gluing the eigenspace to each point in the parameter space creates a complex vector bundle. In order to take into account the Aharonov-Bohm effect that causes a wavefunction's phase to change non-trivially when moving through a magnetic field, we require the vector bundle to be twisted (which manifests as the Chern character of the fiber bundle). Using the Kubo formula, it can be shown that it is precisely this twisting of the vector bundle that creates the quantization of resistance in the quantum hall system.
There are some major problems with this interpretation, but that's why we look at effects like these from more than one point of view. We can also look at the quantum hall effect from the point of view of Laughlin, which emphasizes the physical geometry of the system (which misses the point that the quantization of resistance comes from the structure of the parameter space, but makes evident the need for the localization of states). There is also the point of view of non-commutative geometry, which from what I understand clears up the problems, but it is beyond what I know so I can't explain it :)
I am sorry if that is too much jargon - I do not know how much differential geometry you know. But that is one beautiful example of the application of differential geometry in condensed matter systems. Unfortunately I'm an undergraduate so I have had very little exposure to other examples - I wish I could point you to more! My confidence that there are other examples comes primarily from condensed matter professors telling me so.