{"knowledge_schema": {"broad_category": "Mathematics \u2192 Differential Geometry \u2192 Curvature", "refinement": "This problem involves calculating the Gaussian curvature of a surface defined in polar coordinates using a specific metric.", "specific_scope": "The focus is on a two-dimensional surface described by the given metric in polar coordinates, which requires the application of curvature formulas to derive the Gaussian curvature.", "goal": "Determine the Gaussian curvature of the surface defined by the provided metric."}, "summary": "This problem is centered on the calculation of Gaussian curvature for a surface defined by a specific metric in polar coordinates. The metric given is a function of the radial coordinate and the angular coordinate, and the task involves applying differential geometry principles to compute the curvature. The goal is to find the Gaussian curvature, which in this case is determined to be -1/R^2, indicating a hyperbolic geometry."}