Extension From An Open Subset Of A Manifold To Whole Of The Manifold For Smooth Map Between Two Smooth Manifolds

Introduction

In the realm of differential geometry, manifolds play a crucial role in understanding the properties of spaces that are locally Euclidean. A smooth manifold is a topological space that is locally homeomorphic to Euclidean space, and it is equipped with a smooth atlas, which is a collection of charts that provide a smooth transition between different coordinate systems. In this article, we will discuss the concept of extending a smooth map from an open subset of a manifold to the whole of the manifold, and we will explore the conditions under which such an extension is possible.

Smooth Manifolds and Maps

A smooth manifold is a Hausdorff, second-countable topological space that is locally homeomorphic to Euclidean space. It is equipped with a smooth atlas, which is a collection of charts that provide a smooth transition between different coordinate systems. A smooth map between two smooth manifolds is a map that is smooth in the sense that it is locally given by smooth functions.

Extension of Smooth Maps

Given a smooth map $f: U \to M$ between two smooth manifolds $U$ and $M$, where $U$ is an open subset of $M$, we want to extend $f$ to a smooth map $\tilde{f}: M \to M$. The question is, under what conditions is such an extension possible?

Theorem 1: Extension of Smooth Maps

Let $f: U \to M$ be a smooth map between two smooth manifolds $U$ and $M$, where $U$ is an open subset of $M$. Suppose that $f$ is a diffeomorphism onto its image, and that the image of $f$ is a closed subset of $M$. Then, there exists a smooth map $\tilde{f}: M \to M$ that extends $f$.

Proof

Let $f: U \to M$ be a smooth map between two smooth manifolds $U$ and $M$, where $U$ is an open subset of $M$. Suppose that $f$ is a diffeomorphism onto its image, and that the image of $f$ is a closed subset of $M$. We need to show that there exists a smooth map $\tilde{f}: M \to M$ that extends $f$.

Since $f$ is a diffeomorphism onto its image, we can define a smooth map $\tilde{f}: M \to M$ by setting $\tilde{f}(x) = f(x)$ for all $x \in U$, and $\tilde{f}(x) = x$ for all $x \in M \setminus U$. This map is smooth because it is locally given by smooth functions.

To show that $\tilde{f}$ is a smooth map, we need to show that it is smooth in the sense that it is locally given by smooth functions. Let $x \in M$ be an arbitrary point. If $x \in U$, then $\tilde{f}$ is smooth at $x$ because it is locally given by smooth functions. If $x \in M \setminus U$, then $\tilde{f}$ is smooth at $x$ because it is the identity map.

Therefore, $\tilde{f}$ is a smooth map that extends $f$.

Theorem 2: Extension of Smooth Maps with Compact Support

Let $f: U \to M$ be a smooth map between two smooth manifolds $U$ and $M$, where $U$ is an open subset of $M$. Suppose that $f$ has compact support, and that the image of $f$ is a closed subset of $M$. Then, there exists a smooth map $\tilde{f}: M \to M$ that extends $f$.

Proof

Let $f: U \to M$ be a smooth map between two smooth manifolds $U$ and $M$, where $U$ is an open subset of $M$. Suppose that $f$ has compact support, and that the image of $f$ is a closed subset of $M$. We need to show that there exists a smooth map $\tilde{f}: M \to M$ that extends $f$.

Since $f$ has compact support, we can define a smooth map $\tilde{f}: M \to M$ by setting $\tilde{f}(x) = f(x)$ for all $x \in U$, and $\tilde{f}(x) = 0$ for all $x \in M \setminus U$. This map is smooth because it is locally given by smooth functions.

To show that $\tilde{f}$ is a smooth map, we need to show that it is smooth in the sense that it is locally given by smooth functions. Let $x \in M$ be an arbitrary point. If $x \in U$, then $\tilde{f}$ is smooth at $x$ because it is locally given by smooth functions. If $x \in M \setminus U$, then $\tilde{f}$ is smooth at $x$ because it is the zero map.

Therefore, $\tilde{f}$ is a smooth map that extends $f$.

Conclusion

In this article, we have discussed the concept of extending a smooth map from an open subset of a manifold to the whole of the manifold. We have shown that such an extension is possible under certain conditions, and we have provided two theorems that establish the existence of smooth extensions in different cases. The first theorem establishes the existence of a smooth extension when the map is a diffeomorphism onto its image, and the image of the map is a closed subset of the manifold. The second theorem establishes the existence of a smooth extension when the map has compact support, and the image of the map is a closed subset of the manifold.

References

• Boothby, W. M. (1986). An introduction to differentiable manifolds and Riemannian geometry. Academic Press.
• Spivak, M. (1979). Differential geometry. Publish or Perish.
• Warner, F. W. (1971). Foundations of differentiable manifolds and Lie groups. Springer-Verlag.

Further Reading

• Lee, J. M. (2003). Introduction to smooth manifolds. Springer-Verlag.
• Hirsch, M. W. (1976). Differential topology. Springer-Verlag.
• Milnor, J. W. (1963). Morse theory. Princeton University Press.
  Q&A: Extension from an Open Subset of a Manifold to the Whole of the Manifold for Smooth Map between Two Smooth Manifolds ===========================================================

Q: What is the main goal of extending a smooth map from an open subset of a manifold to the whole of the manifold?

A: The main goal of extending a smooth map from an open subset of a manifold to the whole of the manifold is to create a smooth map that is defined on the entire manifold, rather than just on a subset of it. This can be useful in a variety of applications, such as in the study of differential geometry and topology.

Q: What are the conditions under which a smooth map can be extended from an open subset of a manifold to the whole of the manifold?

A: The conditions under which a smooth map can be extended from an open subset of a manifold to the whole of the manifold depend on the specific properties of the map and the manifold. In general, the map must be a diffeomorphism onto its image, and the image of the map must be a closed subset of the manifold.

Q: What is the significance of the image of the map being a closed subset of the manifold?

A: The image of the map being a closed subset of the manifold is significant because it ensures that the map is well-defined on the entire manifold. If the image of the map is not closed, then the map may not be well-defined on the entire manifold, and it may not be possible to extend the map to the whole of the manifold.

Q: Can a smooth map with compact support be extended from an open subset of a manifold to the whole of the manifold?

A: Yes, a smooth map with compact support can be extended from an open subset of a manifold to the whole of the manifold. In fact, the second theorem that we discussed earlier establishes the existence of a smooth extension in this case.

Q: What are some of the applications of extending a smooth map from an open subset of a manifold to the whole of the manifold?

A: Some of the applications of extending a smooth map from an open subset of a manifold to the whole of the manifold include:

• The study of differential geometry and topology
• The study of Lie groups and Lie algebras
• The study of differential equations and dynamical systems
• The study of geometric analysis and partial differential equations

Q: How can I determine whether a smooth map can be extended from an open subset of a manifold to the whole of the manifold?

A: To determine whether a smooth map can be extended from an open subset of a manifold to the whole of the manifold, you can use the following steps:

1. Check whether the map is a diffeomorphism onto its image.
2. Check whether the image of the map is a closed subset of the manifold.
3. If the map has compact support, check whether the image of the map is a closed subset of the manifold.

If all of these conditions are met, then the map can be extended from an open subset of a manifold to the whole of the manifold.

Q: What are some of the challenges associated with extending a smooth map from an open subset of a manifold to the whole of the manifold?

A: Some of the challenges associated with extending a smooth map from an open subset of a manifold to the whole of the manifold include:

• Ensuring that the map is well-defined on the entire manifold
• Ensuring that the map is smooth on the entire manifold
• Dealing with the possibility that the map may not be well-defined on the entire manifold

Q: How can I overcome the challenges associated with extending a smooth map from an open subset of a manifold to the whole of the manifold?

A: To overcome the challenges associated with extending a smooth map from an open subset of a manifold to the whole of the manifold, you can use the following strategies:

• Use the theorems that we discussed earlier to establish the existence of a smooth extension
• Use the properties of the map and the manifold to ensure that the map is well-defined on the entire manifold
• Use the properties of the map and the manifold to ensure that the map is smooth on the entire manifold

By using these strategies, you can overcome the challenges associated with extending a smooth map from an open subset of a manifold to the whole of the manifold.

Conclusion

In this article, we have discussed the concept of extending a smooth map from an open subset of a manifold to the whole of the manifold. We have established the conditions under which such an extension is possible, and we have provided two theorems that establish the existence of smooth extensions in different cases. We have also discussed some of the applications of extending a smooth map from an open subset of a manifold to the whole of the manifold, and we have provided some strategies for overcoming the challenges associated with this process.

References

• Boothby, W. M. (1986). An introduction to differentiable manifolds and Riemannian geometry. Academic Press.
• Spivak, M. (1979). Differential geometry. Publish or Perish.
• Warner, F. W. (1971). Foundations of differentiable manifolds and Lie groups. Springer-Verlag.

Further Reading

• Lee, J. M. (2003). Introduction to smooth manifolds. Springer-Verlag.
• Hirsch, M. W. (1976). Differential topology. Springer-Verlag.
• Milnor, J. W. (1963). Morse theory. Princeton University Press.