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 map between two smooth manifolds is a fundamental concept that enables us to study the relationships between these spaces. However, when dealing with smooth maps, it is often necessary to extend a map defined on an open subset of a manifold to the entire manifold. This extension process is essential in various areas of mathematics and physics, including differential geometry, topology, and differential equations.

Preliminaries

Before diving into the extension process, let's recall some essential definitions and concepts.

• Smooth Manifold: A smooth manifold is a topological space that is locally Euclidean, meaning that every point in the space has a neighborhood that is homeomorphic to a Euclidean space.
• Smooth Map: A smooth map between two smooth manifolds is a map that is differentiable at every point in the domain manifold.
• Open Subset: An open subset of a manifold is a subset that is open in the manifold's topology.

The Extension Problem

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 $\tilde{f}: M \to M$. This extension process is not always possible, and we need to investigate the conditions under which it is 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 between $U$ and $f(U)$. Then, there exists a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$.

Proof

We can define a smooth map $\tilde{f}: M \to M$ by setting $\tilde{f}(p) = f(p)$ for all $p \in U$ and $\tilde{f}(p) = p$ for all $p \in M \setminus U$. This map is smooth because it is smooth on $U$ and constant on $M \setminus U$.

Theorem 2: Necessary and Sufficient Conditions for Extension

Let $f: U \to M$ be a smooth map between two smooth manifolds $U$ and $M$, where $U$ is an open subset of $M$. Then, the following conditions are necessary and sufficient for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$:

• Condition 1: The map $f$ is a diffeomorphism between $U$ and $f(U)$.
• Condition 2: The map $f$ is injective on $U$.
• Condition 3: The map $f$ is surjective on $U$.

Proof

We need to that these conditions are both necessary and sufficient for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$.

• Necessity: Suppose that there exists a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$. Then, $f$ is a diffeomorphism between $U$ and $f(U)$, and $f$ is injective and surjective on $U$.
• Sufficiency: Suppose that $f$ is a diffeomorphism between $U$ and $f(U)$, and $f$ is injective and surjective on $U$. Then, we can define a smooth map $\tilde{f}: M \to M$ by setting $\tilde{f}(p) = f(p)$ for all $p \in U$ and $\tilde{f}(p) = p$ for all $p \in M \setminus U$. This map is smooth because it is smooth on $U$ and constant on $M \setminus U$.

Conclusion

In this article, we have investigated the extension problem for smooth maps between two smooth manifolds. We have shown that the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ is equivalent to the map $f$ being a diffeomorphism between $U$ and $f(U)$, and $f$ being injective and surjective on $U$. This result has important implications in various areas of mathematics and physics, including differential geometry, topology, and differential equations.

References

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

Further Reading

For further reading on the topic of smooth maps and manifolds, we recommend the following resources:

• Differential Geometry: A comprehensive introduction to differential geometry by Michael Spivak.
• Smooth Manifolds: A detailed treatment of smooth manifolds by John M. Lee.
• Lie Groups: A comprehensive introduction to Lie groups by Anthony W. Knapp.

Open Problems

There are several open problems related to the extension problem for smooth maps between two smooth manifolds. Some of these problems include:

• Problem 1: Can we extend a smooth map $f: U \to M$ to a smooth map $\tilde{f}: M \to M$ even if $f$ is not a diffeomorphism between $U$ and $f(U)$?
• Problem 2: Can we find a necessary and sufficient condition for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ in terms of the topology of the manifolds $U$ and $M$?

Q: What is the extension problem for smooth maps between two smooth manifolds?

A: The extension problem for smooth maps between two smooth manifolds is the problem of extending a smooth map $f: U \to M$ defined on an open subset $U$ of a manifold $M$ to a smooth map $\tilde{f}: M \to M$ defined on the entire manifold $M$.

Q: What are the necessary and sufficient conditions for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$?

A: The necessary and sufficient conditions for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ are:

• Condition 1: The map $f$ is a diffeomorphism between $U$ and $f(U)$.
• Condition 2: The map $f$ is injective on $U$.
• Condition 3: The map $f$ is surjective on $U$.

Q: What is the significance of the extension problem for smooth maps between two smooth manifolds?

A: The extension problem for smooth maps between two smooth manifolds is significant because it has important implications in various areas of mathematics and physics, including differential geometry, topology, and differential equations.

Q: Can we extend a smooth map $f: U \to M$ to a smooth map $\tilde{f}: M \to M$ even if $f$ is not a diffeomorphism between $U$ and $f(U)$?

A: No, we cannot extend a smooth map $f: U \to M$ to a smooth map $\tilde{f}: M \to M$ even if $f$ is not a diffeomorphism between $U$ and $f(U)$. The necessary and sufficient conditions for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ are that $f$ is a diffeomorphism between $U$ and $f(U)$, and $f$ is injective and surjective on $U$.

Q: Can we find a necessary and sufficient condition for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ in terms of the topology of the manifolds $U$ and $M$?

A: No, we cannot find a necessary and sufficient condition for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ in terms of the topology of the manifolds $U$ and $M$. The necessary and sufficient conditions for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ are that $f$ is a diffeomorphism between $U$ and $f(U)$, and $f$ isive and surjective on $U$.

Q: What are some open problems related to the extension problem for smooth maps between two smooth manifolds?

A: Some open problems related to the extension problem for smooth maps between two smooth manifolds include:

• Problem 1: Can we extend a smooth map $f: U \to M$ to a smooth map $\tilde{f}: M \to M$ even if $f$ is not a diffeomorphism between $U$ and $f(U)$?
• Problem 2: Can we find a necessary and sufficient condition for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$ in terms of the topology of the manifolds $U$ and $M$?

Q: What are some resources for further reading on the topic of smooth maps and manifolds?

A: Some resources for further reading on the topic of smooth maps and manifolds include:

• Differential Geometry: A comprehensive introduction to differential geometry by Michael Spivak.
• Smooth Manifolds: A detailed treatment of smooth manifolds by John M. Lee.
• Lie Groups: A comprehensive introduction to Lie groups by Anthony W. Knapp.

Q: What are some areas of mathematics and physics where the extension problem for smooth maps between two smooth manifolds is significant?

A: The extension problem for smooth maps between two smooth manifolds is significant in various areas of mathematics and physics, including:

• Differential Geometry: The extension problem for smooth maps between two smooth manifolds is significant in differential geometry because it has important implications for the study of manifolds and their properties.
• Topology: The extension problem for smooth maps between two smooth manifolds is significant in topology because it has important implications for the study of topological spaces and their properties.
• Differential Equations: The extension problem for smooth maps between two smooth manifolds is significant in differential equations because it has important implications for the study of differential equations and their solutions.

Conclusion

In this Q&A article, we have discussed the extension problem for smooth maps between two smooth manifolds, including the necessary and sufficient conditions for the existence of a smooth map $\tilde{f}: M \to M$ such that $\tilde{f}|_U = f$, and the significance of the extension problem for smooth maps between two smooth manifolds. We have also discussed some open problems related to the extension problem for smooth maps between two smooth manifolds and some resources for further reading on the topic of smooth maps and manifolds.