Submersion

Definition

Submersion refers to a smooth map between manifolds that is a submersion at a point if its differential is surjective at that point. This concept highlights the local behavior of smooth functions and is crucial for understanding how they behave under certain transformations, allowing for the study of critical points and level sets in the context of smooth manifolds.

5 Must Know Facts For Your Next Test

1. A submersion ensures that the image of the map locally resembles the target manifold, facilitating the understanding of topology and geometry.
2. The condition for a map to be a submersion can be tested using the rank theorem, which relates the rank of the differential to the dimensions of the involved manifolds.
3. Submersions preserve dimensionality, meaning if you have a manifold of dimension 'm', its image under a submersion will have dimension equal to 'm' or less.
4. Submersions play a significant role in Morse Theory by allowing one to analyze critical points and their behavior under smooth deformations.
5. Every smooth map can be decomposed into an immersion followed by a submersion, making submersions an essential tool in differential topology.

Review Questions

• How does the concept of submersion relate to critical points in smooth functions?
  • Submersion is essential in understanding critical points because it indicates where a smooth function behaves regularly. At points where a function is a submersion, it implies that there are no critical points nearby since the differential is surjective. This allows mathematicians to classify and analyze these points and their contributions to the topology of the manifold.
• In what ways does the rank theorem help determine whether a smooth map is a submersion?
  • The rank theorem provides a criterion to assess whether a smooth map is a submersion by relating the rank of the differential at a point to the dimensions of the domain and codomain. Specifically, for a map to be classified as a submersion at a point, the rank of its differential must equal the dimension of its target manifold. This theorem aids in identifying regions where submersions occur within smooth maps.
• Evaluate how submersions facilitate analysis in Morse Theory and provide insights into manifold structures.
  • Submersions are vital in Morse Theory because they allow for the study of critical points and their stability under perturbations. By understanding how functions behave locally through submersions, mathematicians can glean insights about the topology of manifolds, such as connectedness and compactness. Additionally, since submersions preserve dimensionality, they help characterize how different regions interact within manifold structures and provide pathways for further exploration in topological features.

Related terms

Immersion: A smooth map between manifolds where the differential is injective at every point, allowing for local embeddings.

Smooth Map: A function between differentiable manifolds that has continuous derivatives of all orders.

Differential: The linear approximation of a function at a given point, representing how the function changes in response to small changes in input.