Let {Xg} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {8}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.

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Given a symmetric tensor on a real vector bundle of dimension two, we construct a space where this tensor corresponds to a scalar function. We prove that under certain regularity conditions such a space and the corresponding scalar function are smooth. We study the topology of this space for the case of surfaces and produce a version of Morse inequalities for symmetric tensors. We apply our results to the geometry of surfaces.

Inspired by the optical phenomenon of conical refraction, discovered by Hamilton in 1832, we study the existence of singular optical phenomena associated with linear differential operators acting on vector fields on a surface. We do this by studying the singularities of the Fresnel hyper-surface associated with the differential operator and show that the existence of these singularities can be accounted for from purely topological considerations. We associate a topological number (an integer) to the space of singularities of the Fresnel hyper-surface, which can be used to “count” the number points at which singular optical phenomena occur.

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