{"knowledge_schema": {"broad_category": "Mathematics \u2192 Differential Geometry \u2192 Curvature", "refinement": "This problem involves calculating the Gaussian curvature of a surface defined by a specific metric in polar coordinates.", "specific_scope": "The metric given is in polar coordinates and describes a two-dimensional surface, where the curvature needs to be determined using the metric components.", "goal": "Calculate the Gaussian curvature of the surface defined by the given metric and identify the correct value from the provided options."}, "summary": "This problem requires understanding the concept of Gaussian curvature in the context of differential geometry. The metric provided in polar coordinates describes a surface, and the task is to compute its Gaussian curvature using the metric components. The Gaussian curvature is a measure of the intrinsic curvature of the surface, and in this case, it has been determined to be -1/R^2, indicating a hyperbolic geometry."}