What is the role of manifold hypothesis in dimensionality reduction?

It assumes high-dimensional data lies on a low-dimensional curved manifold.

The manifold hypothesis states that real-world high-dimensional data (images, speech, text embeddings) lies on or near a much lower-dimensional nonlinear manifold embedded in high-dimensional space. Dimensionality reduction methods aim to recover or approximate this manifold structure, enabling compression while preserving meaningful structure.