  In this paper, we provide a classification of steady solutions to
two-dimensional incompressible Euler equations in terms of the set of flow
angles. The first main result asserts that the set of flow angles of any
bounded steady flow in the whole plane must be the whole circle unless the flow
is a parallel shear flow. In an infinitely long horizontal strip or the upper
half-plane supplemented with slip boundary conditions, besides the two types of
flows appeared in the whole space case, there exists an additional class of
steady flows for which the set of flow angles is either the upper or lower
closed semicircles. This type of flows is proved to be the class of non-shear
flows that have the least total curvature. As consequences, we obtain
Liouville-type theorems for two-dimensional semilinear elliptic equations with
only bounded and measurable nonlinearity, and the structural stability of shear
flows whose all stagnation points are not inflection points, including
Poiseuille flow as a special case. Our proof relies on the analysis of some
quantities related to the curvature of the streamlines.
