  In this paper we study the critical behavior of an $N$-component
${\phi}^{4}$-model in hyperbolic space, which serves as a model of uniform
frustration. We find that this model exhibits a second-order phase transition
with an unusual magnetization texture that results from the lack of global
parallelism in hyperbolic space. Angular defects occur on length scales
comparable to the radius of curvature. This phase transition is governed by a
new strong curvature fixed point that obeys scaling below the upper critical
dimension $d_{uc}=4$. The exponents of this fixed point are given by the
leading order terms of the $1/N$ expansion. In distinction to flat space no
order $1/N$ corrections occur. We conclude that the description of
many-particle systems in hyperbolic space is a promising avenue to investigate
uniform frustration and non-trivial critical behavior within one theoretical
approach.
