Topology Enters Physics

We illustrate Gauss-Bonnet and Poincare-Hopf (GB-PH) theorems in closed two-dimensional geometrical manifolds exhibiting in-plane orientational order which should be spatially homogeneous in equilibrium. Such geometrical shapes inevitable exhibit spatial dependent Gaussian curvature. The latter is a source of local frustrations that could stabilise topological defects (TDs) in the structure. Their presence is in general avoided because they introduce free energy penalties into the system. GB-PH theorem describes total winding number of TDs and indirectly their number in such systems. Namely, due to topological origin the behaviour of TDs obeys conservation laws which are determined by topological invariants. We present applicability of the GB-PH theorem in nematic liquid crystals, where such phenomena could be relatively easily tested experimentally. Furthermore, in terms of these theorems one could also describe quantum Hall effect, via which topology first entered physics.