Submersion

Definition

Submersion is a type of smooth map between differentiable manifolds where the differential of the map is surjective at every point in its domain. This means that for each point in the target manifold, there are points in the source manifold that are mapped to it, allowing for a rich structure in differential topology and the exploration of properties like regular values and smoothness.

5 Must Know Facts For Your Next Test

1. Submersions have the property that their image can locally look like an open subset of the target manifold, making them important for understanding smooth structures.
2. The condition of being a submersion means that if you have a point in your manifold, you can find directions to move such that your map takes you to different directions in the target space.
3. Every submersion is an immersion, but not every immersion is a submersion; this distinction is key when studying mappings between manifolds.
4. In terms of dimension, if you have a submersion from an n-dimensional manifold to an m-dimensional manifold, then n must be greater than or equal to m.
5. Submersions play a crucial role in defining fiber bundles and studying their properties, as they help establish connections between different manifolds.

Review Questions

• How does a submersion relate to regular values, and why is this relationship important in differential topology?
  • A submersion guarantees that for every regular value in the target manifold, the preimage consists entirely of regular points. This relationship is significant because it allows us to analyze the topology of manifolds using submersions. The ability to identify regular values helps in constructing level sets and understanding how these sets behave under smooth maps, which is essential for studying differential topology.
• Discuss the role of the differential of a map in determining whether a given smooth function is a submersion.
  • The differential of a smooth map plays a critical role in classifying that map as a submersion. For a smooth function to be considered a submersion at a given point, its differential must be surjective at that point. This means that it must cover all possible directions in the target space, ensuring that small changes in the input produce substantial changes in the output. Understanding this relationship allows us to determine local behavior and properties of maps between manifolds.
• Evaluate how understanding submersions can influence our comprehension of fiber bundles and their applications.
  • Understanding submersions significantly enhances our comprehension of fiber bundles since they provide essential insights into how fibers over a base manifold are structured. By analyzing how a submersion connects different manifolds, we can explore how fibers behave and interact under smooth mappings. This evaluation opens doors to applications in various fields such as physics and engineering, where fiber bundles model complex systems and structures. The capacity to analyze changes within these bundles relies heavily on our grasp of submersions.

Related terms

Regular Value: A regular value is a point in the target manifold such that all points in the preimage are regular points, meaning the differential of the map is surjective at those points.

Smooth Map: A smooth map is a function between differentiable manifolds that has continuous derivatives of all orders, preserving the structure and properties of the manifolds.

Differential of a Map: The differential of a map is a linear approximation of the map at a point, which captures how the map changes near that point and is crucial for analyzing the local behavior of the map.