Maps of manifolds

In the context of differential geometry and topology, "maps of manifolds" typically refers to smooth or continuous functions that associate points from one manifold to another. Manifolds themselves are mathematical structures that generalize the concept of curves and surfaces to higher dimensions. They can be thought of as "locally Euclidean" spaces, meaning that around any point in a manifold, one can find a neighborhood that looks like Euclidean space.

A **Riemannian submersion** is a specific type of mathematical structure that arises in differential geometry. It involves two Riemannian manifolds and a smooth map between them that preserves certain geometric properties. More formally, let \( (M, g_M) \) and \( (N, g_N) \) be two Riemannian manifolds, where \( g_M \) and \( g_N \) are their respective Riemannian metrics.

In mathematics, particularly in the fields of topology and differential geometry, the term "submersion" refers to a specific type of smooth map between differentiable manifolds. A smooth map \( f: M \to N \) is called a submersion at a point \( p \in M \) if its differential \( df_p: T_pM \to T_{f(p)}N \) is surjective.