Riemannian manifolds are a special type of smooth manifold equipped with a Riemannian metric, which allows for the measurement of distances and angles on the manifold. This structure gives the manifold a way to study geometric properties, enabling concepts like curvature and geodesics, which are crucial for understanding how the manifold behaves. Riemannian manifolds form the foundation for many important areas in mathematics, such as differential geometry and general relativity.
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