  Non-Euclidean geometry, discovered by negating Euclid's parallel postulate,
has been of considerable interest in mathematics and related fields for the
description of geographical coordinates, Internet infrastructures, and the
general theory of relativity. Notably, an infinite number of regular
tessellations in hyperbolic geometry-hyperbolic lattices-can extend Euclidean
Bravais lattices and the consequent band theory to non-Euclidean geometry. Here
we demonstrate topological phenomena in hyperbolic geometry, exploring how the
quantized curvature and edge dominance of the geometry affect topological
phases. We report a recipe for the construction of a Euclidean photonic
platform that inherits the topological band properties of a hyperbolic lattice
under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean
analogue of the quantum spin Hall effect. For hyperbolic lattices with
different quantized curvatures, we examine the topological protection of
helical edge states and generalize Hofstadter's butterfly, showing the unique
spectral sensitivity of topological immunity in highly curved hyperbolic
planes. Our approach is applicable to general non-Euclidean geometry and
enables the exploitation of infinite lattice degrees of freedom for band
theory.
