A Riemannian manifold is a type of mathematical space that combines the concepts of geometry and calculus. It is a smooth, curved surface where each point has a defined way to measure distances and angles. This is achieved through a special tool called a Riemannian metric, which assigns a positive-definite inner product to the tangent space at each point. Riemannian manifolds are essential in various fields, including general relativity, where they describe the curvature of space-time. They also play a crucial role in differential geometry and topology, helping mathematicians understand complex shapes and their properties.