three-dimensional manifolds

A three-dimensional manifold is a space that at every small region looks like ordinary 3D space around us, similar to how a globe’s surface appears flat when viewed up close. Think of it as a shape or universe that can be complex globally, but locally resembles familiar three-dimensional space. Examples include our universe itself or more abstract shapes like a 3D torus (doughnut-shaped universe). These manifolds help mathematicians study possible shapes of space, understand the universe’s structure, and explore geometrical properties, all while ensuring each tiny neighborhood is geometrically simple and understandable.