  In this paper, we provide new and simpler proofs of two theorems of Gluck and
Harrison on contact structures induced by great circle or line fibrations.
Furthermore, we prove that a geodesic vector field whose Jacobi tensor is
parallel along flow lines (e.g. if the underlying manifold is locally
symmetric) induces a contact structure if the 'mixed' sectional curvatures are
nonnegative, and if a certain nondegeneracy condition holds. Additionally, we
prove that in dimension three, contact structures admitting a Reeb flow which
is either periodic, isometric, or free and proper, must be universally tight.
In particular, we generalise an earlier result of Geiges and the author, by
showing that every contact form on $\mathbb{R}^3$ whose Reeb vector field spans
a line fibration is necessarily tight. Furthermore, we provide a
characterisation of isometric Reeb vector fields. As an application, we recover
a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector
fields, and we classify closed contact $3$-manifolds with isometric Reeb flows
(also known as $R$-contact manifolds) up to diffeomorphism.
