Immersed submanifold

Definition

An immersed submanifold is a subset of a manifold that has a differentiable structure and can be locally represented as a differentiable map from a Euclidean space into the larger manifold. It retains some properties of submanifolds, but unlike embedded submanifolds, it may self-intersect and does not necessarily have a topology that matches the ambient manifold in every point. This concept connects with smooth maps, which describe how these structures can be smoothly related to one another.

5 Must Know Facts For Your Next Test

1. An immersed submanifold can be defined by an immersion, which is a smooth map whose differential is injective at every point.
2. While immersed submanifolds allow for self-intersections, they still maintain local homeomorphism to Euclidean spaces.
3. The concept of immersions is crucial in understanding the behavior of curves and surfaces within higher-dimensional spaces.
4. In contrast to embedded submanifolds, immersed submanifolds do not necessarily have the same global topology as their ambient manifold.
5. The study of immersed submanifolds often involves examining properties like their tangent spaces and how they interact with smooth maps.

Review Questions

• How does the definition of an immersed submanifold differ from that of an embedded submanifold?
  • An immersed submanifold allows for self-intersections and does not require its topology to match that of the ambient manifold globally. In contrast, an embedded submanifold must be homeomorphic to an open subset of its ambient space and cannot self-intersect. This distinction is significant because it affects how we can utilize concepts like tangent spaces and smooth maps when studying these structures.
• Discuss the importance of immersions in understanding the behavior of curves within higher-dimensional manifolds.
  • Immersions provide critical insight into how curves behave in higher-dimensional spaces by allowing them to be represented as smooth maps. This representation helps us understand local properties like tangents and curvature while accommodating self-intersections. The ability to analyze curves as immersed submanifolds lets us apply differential geometric techniques to explore their characteristics without being restricted by the rigid requirements of embeddings.
• Evaluate how the concept of an immersed submanifold enhances our understanding of smooth maps and their relationships between different manifolds.
  • The concept of an immersed submanifold enriches our understanding of smooth maps by illustrating how these maps can relate distinct manifolds while allowing for complex interactions like self-intersections. Immersed submanifolds showcase scenarios where the standard rules governing embeddings do not apply, prompting deeper investigation into their properties. This examination highlights the versatility and richness of differentiable structures, enabling mathematicians to develop more comprehensive models that account for both local and global behaviors in manifold theory.

Related terms

differentiable structure: A differentiable structure is a collection of charts that allows for the definition of differentiability on a manifold, providing a way to perform calculus on it.

tangent space: The tangent space at a point on a manifold consists of all possible directions in which one can tangentially pass through that point, forming a vector space that plays a key role in differential geometry.

embedding: An embedding is a type of smooth map that preserves the manifold's structure and allows an immersed submanifold to be considered as a subspace of the ambient manifold without self-intersection.