An embedded submanifold is a subset of a manifold that inherits its manifold structure through a smooth embedding, meaning it can be treated as a manifold in its own right while being contained within a larger manifold. This concept plays a crucial role in understanding the relationship between different manifolds, particularly in how they can share geometric and topological properties while maintaining distinct identities.
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An embedded submanifold can be characterized by the existence of a smooth embedding from an open set in Euclidean space into the larger manifold.
The topology of an embedded submanifold is induced from the larger manifold, making it possible to study its properties using the tools developed for the larger space.
Embedded submanifolds can have different dimensions than the ambient manifold; for example, a circle (1-dimensional) can be an embedded submanifold of a 2-dimensional sphere.
Every embedded submanifold is also a regular submanifold, but not all submanifolds are necessarily embedded; some may have singularities or not maintain a smooth structure.
Examples of embedded submanifolds include curves and surfaces in higher-dimensional spaces, such as lines in $ extbf{R}^3$ or spheres in $ extbf{R}^4$.
Review Questions
How does the concept of smooth embedding help establish the relationship between an embedded submanifold and its ambient manifold?
Smooth embedding allows an embedded submanifold to inherit properties from its ambient manifold while maintaining its own distinct manifold structure. Through smooth embeddings, one can apply tools of differential geometry and topology to study the embedded submanifold without losing sight of how it relates to the larger space. The immersion property ensures that local features are preserved, helping to understand both global and local behaviors.
In what ways do embedded submanifolds differ from regular submanifolds, and why is this distinction important?
The distinction between embedded submanifolds and regular submanifolds lies in the preservation of the smooth structure and the avoidance of singularities. Embedded submanifolds can be smoothly mapped into their ambient spaces without loss of differentiability, whereas regular submanifolds may have points where this is not true. Understanding these differences is crucial for applications in geometry and physics, where properties like curvature and topology must be carefully analyzed.
Evaluate the significance of examples like circles in $ extbf{R}^3$ or spheres in $ extbf{R}^4$ as embedded submanifolds within their respective ambient manifolds.
Examples such as circles in $ extbf{R}^3$ or spheres in $ extbf{R}^4$ illustrate how low-dimensional structures can fit neatly into higher-dimensional spaces while retaining their own unique geometrical characteristics. This embedding showcases how we can study complex shapes by analyzing simpler forms within them, enhancing our understanding of global geometric properties through local analysis. Such examples are fundamental in applications ranging from theoretical physics to advanced geometric studies, demonstrating how different dimensionalities interact.
Related terms
Smooth Embedding: A smooth embedding is a differentiable map from one manifold to another that is both an immersion and a homeomorphism onto its image, allowing the embedded submanifold to retain the smooth structure.
A submanifold is a subset of a manifold that has its own manifold structure, potentially with different dimensionality, but does not necessarily have to be embedded within another manifold.
An immersion is a differentiable function between manifolds that is locally injective and its differential is injective at every point, allowing for the preservation of manifold properties in a more general sense.