Regular Point Theorem of Manifolds with Boundaries

In summary, the regular value theorem in most textbooks for manifolds without boundaries states that the preimage of a regular value is an embedding submanifold. However, it is worth considering how this applies to manifolds with boundaries. Milnor's book, "Topology from the Differentiable Viewpoint" is recommended as it covers this topic and others in Differential Topology. It is suggested to extend the regular value theorem to manifolds with boundaries as a good exercise.
  • #1
Fangyang Tian
17
0
Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries??
Many Thanks!
 
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  • #2
Fangyang Tian said:
Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries??
Many Thanks!

I passionately recommend Milnor's book, Topology from the Differentiable Viewpoint which covers this and many other topics in Differential Topology.

The regular value theorem is an application of the Implicit Function Theorem. Extending it to a manifold with boundary is a good exercise.
 

FAQ: Regular Point Theorem of Manifolds with Boundaries

What is the Regular Point Theorem of Manifolds with Boundaries?

The Regular Point Theorem of Manifolds with Boundaries is a fundamental result in topology that states that every point on a manifold with boundaries has a neighborhood that is homeomorphic to an open subset of a Euclidean space. This means that manifolds with boundaries are locally similar to flat, Euclidean spaces.

How is the Regular Point Theorem of Manifolds with Boundaries used in mathematics?

The Regular Point Theorem of Manifolds with Boundaries is used in many areas of mathematics, including differential geometry, topology, and mathematical physics. It is a key tool for studying the local properties of manifolds with boundaries and is often used to prove other theorems in these fields.

Can the Regular Point Theorem of Manifolds with Boundaries be extended to higher dimensions?

Yes, the Regular Point Theorem of Manifolds with Boundaries can be extended to higher dimensions. In fact, the theorem is a special case of a more general result known as the Whitney Embedding Theorem, which states that any smooth manifold can be embedded in a Euclidean space of sufficiently high dimension.

Are there any applications of the Regular Point Theorem of Manifolds with Boundaries in real-world problems?

Yes, the Regular Point Theorem of Manifolds with Boundaries has many applications in real-world problems, particularly in physics and engineering. It is used to model and analyze various physical systems, such as fluid dynamics, electromagnetism, and general relativity.

What are the implications of the Regular Point Theorem of Manifolds with Boundaries for understanding the topology of space?

The Regular Point Theorem of Manifolds with Boundaries is an important result for understanding the topology of space. It tells us that, locally, space can be approximated by a flat, Euclidean space. This has implications for our understanding of the global structure of space and how it can be deformed and distorted without changing its local properties.

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