Shared autonomy (SA), which combines user inputs with autonomous capabilities, presents a significant opportunity for assistive robotics. A key challenge in SA is the dimensionality gap: the mismatch between low-dimensional user inputs from familiar interfaces (e.g., 2D joysticks) and the high-dimensional control required by robot manipulators. To enhance usability and acceptance, this mapping must be as simple and intuitive as possible. We introduce GeoSACS, a geometric framework for SA. GeoSACS uses canal surfaces to encode task structure with as few as two demonstrations. While the robot moves autonomously along the canal, users can then make corrections on the 2D planar circular cross-sections orthogonal to the robot motion. By leveraging geometric structure to partition the 6D control space between the robot and the user, GeoSACS allows the intuitive mapping of 2D user inputs to 6D end-effector control. We describe GeoSACS and evaluate its underlying assumptions in a user study against two baselines. Results from the study demonstrate reduced workload and improved performance, providing insights for the design of future SA systems.