Next Point-of-Interest (POI) recommendation is a critical task in modern Location-Based Social Networks (LBSNs), aiming to model the complex decision-making process of human mobility to provide personalized recommendations for a user's next check-in location. Existing hyperbolic POI recommendation models, predominantly based on rotations and graph representations, have been extensively investigated. Although hyperbolic geometry has proven superior in representing hierarchical data with low distortion, current hyperbolic sequence models typically rely on performing recurrence via expensive Möbius operations directly on the manifold. This incurs prohibitive computational costs and numerical instability, rendering them ill-suited for trajectory modeling. To resolve this conflict between geometric representational power and sequential efficiency, we propose GTR-Mamba, a novel framework featuring Geometry-to-Tangent Routing. GTR-Mamba strategically routes complex state transitions to the computationally efficient Euclidean tangent space. Crucially, instead of a static approximation, we introduce a Parallel Transport (PT) mechanism that dynamically aligns tangent spaces along the trajectory. This ensures geometric consistency across recursive updates, effectively bridging the gap between the curved manifold and linear tangent operations. This process is orchestrated by an exogenous spatio-temporal channel, which explicitly modulates the SSM discretization parameters. Extensive experiments on three real-world datasets demonstrate that GTR-Mamba consistently outperforms state-of-the-art baselines in next POI recommendation.