Whitney Embedding Theorem

Definition

The Whitney Embedding Theorem states that any smooth manifold can be embedded into a Euclidean space of sufficiently high dimension. This theorem is pivotal in the study of topology and differential geometry, as it establishes the foundational relationship between smooth manifolds and Euclidean spaces, highlighting the importance of embeddings in understanding manifold structure.

5 Must Know Facts For Your Next Test

1. The theorem applies specifically to smooth manifolds and guarantees an embedding in $ ext{R}^n$ for an appropriate choice of $n$, typically larger than or equal to twice the dimension of the manifold plus one.
2. The Whitney Embedding Theorem is crucial for proving other important results in topology and geometry, such as the existence of immersion and the study of characteristic classes.
3. An embedding as stated in the theorem is always a smooth injective map that retains the manifold's structure when viewed within Euclidean space.
4. The theorem emphasizes that while every smooth manifold can be embedded, not all topological spaces have this property, showcasing a key distinction between topological and smooth categories.
5. The theorem relies on foundational concepts from differential topology, particularly the use of partitions of unity and the properties of tangent spaces.

Review Questions

• How does the Whitney Embedding Theorem demonstrate the relationship between smooth manifolds and Euclidean spaces?
  • The Whitney Embedding Theorem shows that any smooth manifold can be smoothly embedded into a sufficiently high-dimensional Euclidean space. This relationship is essential because it allows us to utilize familiar Euclidean geometry techniques to study complex structures of manifolds. By embedding a manifold in $ ext{R}^n$, we gain insights into its properties and behaviors, which can be difficult to analyze in their original topological form.
• Discuss the implications of the Whitney Embedding Theorem for the study of immersions and how it connects to other results in differential topology.
  • The Whitney Embedding Theorem implies that since every smooth manifold can be embedded, it must also admit an immersion into some Euclidean space. This connection is significant because immersions preserve many geometric properties even if they do not guarantee an embedding due to potential overlaps. The theorem serves as a foundation for other results in differential topology, linking the concepts of embeddings, immersions, and the overall behavior of manifolds under continuous transformations.
• Evaluate how the Whitney Embedding Theorem differentiates between smooth manifolds and general topological spaces, particularly concerning their embedding properties.
  • The Whitney Embedding Theorem clearly illustrates that while every smooth manifold has the ability to be embedded in Euclidean space, not all topological spaces share this characteristic. This distinction highlights a key aspect of differential topology: embeddings are closely related to smooth structures rather than just topological ones. Many topological spaces cannot be embedded in any Euclidean space without losing some structural integrity, showcasing how embeddings are a powerful tool for understanding manifolds specifically within a differentiable framework.

Related terms

Smooth Manifold: A smooth manifold is a topological manifold equipped with a globally defined differential structure that allows for the differentiation of functions.

Embedding: An embedding is a type of function that allows one mathematical object to be represented within another, preserving its structure and properties.

Euclidean Space: Euclidean space is a fundamental geometric space characterized by its flatness and defined by familiar concepts such as points, lines, and planes.