Submersion

Definition

Submersion is a type of smooth map between manifolds where the differential at every point is surjective. This means that the map takes tangent vectors from the domain and covers all possible directions in the tangent space of the target manifold. The concept is important for understanding how different manifolds relate to each other and provides insight into the structure of their tangent spaces.

5 Must Know Facts For Your Next Test

1. In submersion, since the differential is surjective, the image of the map can have a dimension equal to or greater than that of the target manifold.
2. Submersions are important for defining fibred structures, as they allow for the construction of local coordinates on the target manifold.
3. Every immersion is locally a submersion if it has full rank in its differential, allowing for certain smooth structures to be connected.
4. A submersion ensures that for every point in the target manifold, there exists at least one point in the source manifold mapping to it.
5. The existence of local coordinates around points in both manifolds plays a crucial role in ensuring the properties and applications of submersions.

Review Questions

• How does a submersion differ from an immersion, and what implications does this difference have on their respective tangent spaces?
  • A submersion differs from an immersion primarily in terms of the behavior of their differentials. In an immersion, the differential is injective, meaning it maintains dimensionality without collapsing directions. Conversely, in a submersion, the differential is surjective, covering all directions in the tangent space of the target manifold. This means that while immersions allow for distinct embedding of manifolds, submersions facilitate a broader mapping that captures more global structure.
• Discuss the significance of surjectivity in the differential of a submersion and its impact on local coordinates.
  • The surjectivity of the differential in a submersion is significant because it guarantees that every tangent vector in the target manifold can be represented by at least one tangent vector in the source manifold. This property allows for local coordinates to be defined around points on both manifolds, ensuring that we can understand how they relate and interact. Additionally, this relationship makes it easier to analyze complex structures and transitions between different manifolds.
• Evaluate how submersions contribute to our understanding of manifold theory and its applications across various fields such as physics and engineering.
  • Submersions contribute greatly to manifold theory by providing a way to connect different manifolds while preserving their structures through smooth maps. Their surjective nature allows for diverse applications such as analyzing phase spaces in physics or designing systems in engineering where different states must relate smoothly. By ensuring that mappings cover all necessary dimensions and directions, submersions enable mathematicians and scientists to model complex phenomena accurately and cohesively across disciplines.

Related terms

Immersion:

A smooth map between manifolds where the differential is injective, meaning it preserves the dimension of the manifold being mapped.

Differential:

A linear map that approximates the change of a function between manifolds at a given point, capturing how the function behaves locally.

Tangent Space:

The vector space consisting of all tangent vectors at a point on a manifold, representing all possible directions in which one can move from that point.