  We relate the geometry of curves to the notion of hyperbolicity in real
algebraic geometry. A hyperbolic variety is a real algebraic variety that (in
particular) admits a real fibered morphism to a projective space whose
dimension is equal to the dimension of the variety. We study hyperbolic
varieties with a special interest in the case of hypersurfaces that admit a
real algebraic ruling. The central part of the paper is concerned with secant
varieties of real algebraic curves where the real locus has the maximal number
of connected components, which is determined by the genus of the curve. For
elliptic normal curves, we further obtain definite symmetric determinantal
representations for the hyperbolic secant hypersurfaces, which implies the
existence of symmetric Ulrich sheaves of rank one on these hypersurfaces.
