Immersion

Definition

An immersion is a smooth map between differentiable manifolds that reflects the local structure of the manifolds, allowing for the differential structure to be preserved. This means that at each point in the domain, the map can be represented by a differentiable function whose derivative is injective, indicating that locally, the manifold can be thought of as being 'inserted' into another manifold without self-intersections.

5 Must Know Facts For Your Next Test

1. An immersion is defined by the requirement that its differential is injective at every point in the domain, meaning it does not collapse dimensions locally.
2. Every embedding is an immersion, but not every immersion is an embedding; immersions may self-intersect while embeddings do not.
3. The concept of immersions can be used to study various types of manifolds, including spheres and tori, revealing their complex structure through local coordinates.
4. In applications related to differential topology, immersions are crucial in understanding how manifolds can be transformed and related to each other.
5. Immersions play a significant role in transversality theory, where they are used to analyze intersections and the behavior of maps between manifolds.

Review Questions

• How does the definition of immersion relate to the concept of smooth maps between differentiable manifolds?
  • An immersion is a type of smooth map between differentiable manifolds where the differential at each point is injective. This ensures that the local behavior of the manifold's structure is preserved under the map. Smoothness allows for differentiability in terms of calculus on manifolds, meaning we can analyze how points from one manifold correspond to another while maintaining essential geometric properties.
• Discuss the differences between immersions and embeddings and why these differences matter in differential topology.
  • While both immersions and embeddings are smooth maps between manifolds, an immersion allows for local injectivity but may self-intersect, whereas an embedding requires both local injectivity and global injectivity (no self-intersections). This distinction is important because embeddings preserve more structural information about the manifold's geometry. Understanding this difference helps in studying various applications in topology where the preservation of structure is crucial.
• Evaluate how immersions contribute to the understanding of transversality in differential topology and their implications for manifold intersections.
  • Immersions play a vital role in transversality theory because they allow for the examination of how two or more manifolds intersect when mapped through smooth functions. By using immersions, one can show conditions under which the intersections occur cleanly (transversally) without tangential overlaps. This understanding leads to powerful results in topology that describe how complex shapes and spaces interact, impacting both theoretical frameworks and practical applications in geometry.

Related terms

Differential Structure:

A framework on a manifold that allows for the definition of concepts like smoothness and differentiability, essential for studying how functions behave on manifolds.

Tangent Space:

The vector space consisting of all tangent vectors at a point on a manifold, providing a way to analyze local properties of curves and surfaces.

Submanifold:

A subset of a manifold that is itself a manifold, where the inclusion map is an immersion, allowing us to study lower-dimensional structures within higher-dimensional spaces.