In this thesis the author develops techniques to study contact structures via Riemannian geometry. The main observation is a relation between characteristic surfaces of contact structures and zero sets of solutions to certain subelliptic PDEs. This relation makes it possible to derive, under a symmetry assumption, necessary and sufficient conditions for tightness of contact structures arising from a certain class of invariant curl eigenfields. Further, it has implications in the energy relaxation of this special class of fluid flows. Specifically, the author shows existence of an energy minimizing curl eigenfield which is orthogonal to an overtwisted contact structure. It provides a counterexample to the conjecture of Etnyre and Ghrist posed in their work on hydrodynamics of contact structures.