Submersion and Immersion

Definition

Submersion and immersion refer to types of smooth mappings between manifolds, focusing on the relationship between their dimensions. Specifically, a submersion is a smooth map where the differential is surjective, while an immersion is a smooth map where the differential is injective. These concepts are crucial for understanding how coordinate charts and atlases describe manifold structures and how they interact with each other through smooth transitions.

5 Must Know Facts For Your Next Test

1. In a submersion, the image of the differential covers the entire tangent space of the target manifold at every point in its domain.
2. An immersion allows for locally defined coordinate charts that preserve angles but may not necessarily preserve lengths.
3. A smooth map can be both an immersion and a submersion, which is known as a submersion-immersion or a diffeomorphism if it has a smooth inverse.
4. Submersions help in defining local coordinates on manifolds by providing a way to analyze maps from lower-dimensional spaces to higher-dimensional ones.
5. Immersions can result in self-intersecting images, whereas submersions are more controlled in their mapping properties.

Review Questions

• How do submersions and immersions relate to the structure of coordinate charts on manifolds?
  • Submersions and immersions are essential in defining how coordinate charts relate to each other on manifolds. Submersions provide a way to transition from one chart to another by ensuring that the tangent spaces align correctly, allowing for smooth changes in coordinates. Immersions, on the other hand, enable local representations of manifolds while preserving certain properties of the space, which is important for understanding how charts overlap and interact.
• Discuss the implications of a smooth map being both an immersion and a submersion. What does this mean for the mapping between manifolds?
  • When a smooth map is both an immersion and a submersion, it indicates that the mapping is particularly well-behaved, forming a diffeomorphism between manifolds. This means that not only does the map locally preserve the manifold's structure through injectivity, but it also covers every point in the target manifold fully through surjectivity. This quality is crucial for establishing equivalences between different manifold representations and for defining global properties.
• Evaluate how submersions and immersions facilitate our understanding of differentiable structures on manifolds and their applications in advanced geometry.
  • Submersions and immersions play critical roles in enhancing our comprehension of differentiable structures by illustrating how manifolds can be mapped smoothly onto one another. This understanding leads to applications such as defining tangent bundles and exploring curvature within advanced geometric frameworks. By analyzing these mappings, we can derive significant insights into the behavior of geometric objects under various transformations, which is fundamental for fields such as physics, robotics, and computer graphics.

Related terms

Differential: The differential of a smooth map is a linear map that describes how the map behaves near a point, representing the best linear approximation to the map at that point.

Smooth Map:

A smooth map is a function between manifolds that has continuous derivatives of all orders, ensuring the differentiable structure is preserved in the mapping.

Manifold:

A manifold is a topological space that locally resembles Euclidean space, allowing for the application of calculus and differential geometry.