  Let $H$ be either a complex inner product space of dimension at least two, or
a real inner product space of dimension at least three. Let us fix an
$\alpha\in \left(0,\tfrac{\pi}{2}\right)$. The purpose of this paper is to
characterize all bijective transformations on the projective space $P(H)$
obtained from $H$ which preserves the angle $\alpha$ between lines in both
directions. (We emphasize that we do not assume anything about other angles).
For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every
$\alpha$, and when $H$ is a complex inner product space of dimension at least
three we describe the structure of these transformations for
$\alpha\leq\tfrac{\pi}{4}$. As an application, we give an Uhlhorn-type
generalization of a famous theorem of Wigner which is considered to be a
cornerstone of the mathematical foundations of quantum mechanics. Namely, we
show that under the above assumptions, every bijective map on the set of pure
states of a quantum mechanical system that preserves the transition probability
$\cos^2\alpha$ in both directions is a Wigner symmetry (i.e. it automatically
preserves all transition probability), except for the case when
$H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi}{4}$ where an additional possibility
occurs. We note that the classical theorem of Uhlhorn is the solution for the
$\alpha = \tfrac{\pi}{2}$ case. Usually in the literature, results which are
connected to Wigner's theorem are discussed under the assumption of
completeness of $H$, however, here we shall remove this unnecessary hypothesis.
Our main tools are a characterization of bijective maps on unit spheres of real
inner product spaces which preserve an angle in both directions, and an
extension of Uhlhorn's theorem for non-complete inner product spaces.
