The spin-splitting symmetries of altermagnets, such as $d, g$, and $i$ wave, are strictly constrained by the spin point group symmetries of their underlying Euclidean crystal lattices. This raises a fundamental question: Can we transcend this crystallographic constraint to create entirely new magnetic symmetries by manipulating the intrinsic geometry of space? Here, we develop a general theoretical framework for introducing altermagnetic order on hyperbolic Bravais lattices, which are characterized by negative curvature. We discover spin-splitting patterns with exceptionally high-fold symmetries, such as fourfold $g$ wave, 10-fold $m$ wave, and even higher orders. The symmetries of these patterns are directly dictated by the non-Abelian Fuchsian groups describing hyperbolic translations and are unattainable in Euclidean space. Furthermore, we uncover an unexpected spin degeneracy in hyperbolic altermagnets. This degeneracy is not protected by symmetry but originates from flat bands induced by the unique connectivity of the hyperbolic lattice, which arise from the local destructive interference of wave functions. This work not only extends the concept of altermagnetism from flat to curved space but also opens an avenue for designing spin phases through geometry engineering, providing a theoretical platform to explore the profound connections among geometry, magnetism, and electronic properties.

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• Altermagnetism
• Electronic structure of atoms & molecules
• Altermagnets
• Hyperbolic lattice
• Cluster methods