A differentiable manifold is a mathematical structure that generalizes the concept of curves and surfaces to higher dimensions. It consists of a set of points that locally resemble Euclidean space, allowing for the definition of calculus concepts like derivatives. This property enables the study of smooth shapes and their properties. Differentiable manifolds are essential in various fields, including physics, where they provide the framework for general relativity, and in geometry, where they help understand complex shapes. They are characterized by charts and atlases, which facilitate the transition between different local coordinate systems.