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 coordinate system on the manifold. When we have two smooth manifolds, we often need to consider smooth maps between them, which are maps that are smooth in the sense that they can be locally represented by smooth functions. In this article, we will discuss the extension of a smooth map from an open subset of one manifold to the whole of the manifold, and we will explore the conditions under which such an extension is possible.

Smooth Maps between Manifolds

A smooth map between two smooth manifolds is a map that is smooth in the sense that it can be locally represented by smooth functions. More formally, let $M$ and $N$ be two smooth manifolds, and let $U \subset M$ be an open subset of $M$. A smooth map $f: U \to N$ is a map that is smooth in the sense that for every point $p \in U$, there exists a chart $(U_p, \phi_p)$ on $M$ and a chart $(V_p, \psi_p)$ on $N$ such that $f(U_p \cap U) \subset V_p$ and the map $\psi_p \circ f \circ \phi_p^{-1}: \phi_p(U_p \cap U) \to \psi_p(V_p)$ is smooth.

Extension of Smooth Maps

Given a smooth map $f: U \to N$ between two smooth manifolds $M$ and $N$, we would like to extend $f$ to a smooth map from the whole of $M$ to $N$. This means that we would like to find a smooth map $\tilde{f}: M \to N$ such that $\tilde{f}|_U = f$. In other words, we would like to extend the domain of $f$ from the open subset $U$ to the whole of $M$.

Existence of Extensions

The existence of an extension of a smooth map from an open subset of one manifold to the whole of the manifold depends on the properties of the map and the manifolds involved. In general, if the map is smooth and the manifolds are compact, then an extension exists. However, if the map is not smooth or the manifolds are not compact, then an extension may not exist.

Conditions for Existence

There are several conditions that must be satisfied in order for an extension of a smooth map from an open subset of one manifold to the whole of the manifold to exist. These conditions include:

• Smoothness: The map must be smooth on the open subset $U$.
• Compactness: The manifolds $M$ and $N$ must be compact.
• Connectedness: The manifolds $M$ and $N$ must be connected.
• Local triviality: The map must be locally trivial, meaning that it can be by a smooth function on a neighborhood of each point in $U$.

Construction of Extensions

If an extension of a smooth map from an open subset of one manifold to the whole of the manifold exists, then it can be constructed using several different methods. These methods include:

• Partition of unity: A partition of unity is a collection of smooth functions that are positive on a neighborhood of each point in the manifold and sum to 1 on the whole manifold. Using a partition of unity, we can construct an extension of the map by multiplying the map by a smooth function that is 1 on the open subset $U$ and 0 outside of $U$.
• Smooth approximation: A smooth approximation of a map is a smooth map that is close to the original map. Using a smooth approximation, we can construct an extension of the map by approximating the map on a neighborhood of each point in $U$ and then extending the approximation to the whole manifold.
• Local trivialization: A local trivialization of a map is a smooth map that represents the map on a neighborhood of each point in $U$. Using a local trivialization, we can construct an extension of the map by representing the map on a neighborhood of each point in $U$ and then extending the representation to the whole manifold.

Examples

There are several examples of smooth maps between manifolds that can be extended to the whole manifold. These examples include:

• Identity map: The identity map on a manifold is a smooth map that maps each point to itself. The identity map can be extended to the whole manifold using a partition of unity.
• Constant map: A constant map on a manifold is a smooth map that maps each point to a fixed point. The constant map can be extended to the whole manifold using a smooth approximation.
• Smooth embedding: A smooth embedding of a manifold into another manifold is a smooth map that is injective and has a smooth inverse. A smooth embedding can be extended to the whole manifold using a local trivialization.

Conclusion

In conclusion, the extension of a smooth map from an open subset of one manifold to the whole of the manifold is a fundamental problem in differential geometry. The existence of an extension depends on the properties of the map and the manifolds involved, and several conditions must be satisfied in order for an extension to exist. Several methods can be used to construct an extension of a smooth map, including partition of unity, smooth approximation, and local trivialization. Examples of smooth maps that can be extended to the whole manifold include the identity map, constant map, and smooth embedding.

References

• Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry. Academic Press.
• Hirsch, M. W. (1976). Differential topology. Springer-Verlag.
• Spivak, M. (1979). A comprehensive introduction to differential geometry. Publish or Perish.
  Q&A: Extension of Smooth Maps between Manifolds =============================================

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

A: The main goal of extending a smooth map from an open subset of one manifold to the whole of the manifold is to find a smooth map that represents the original map on the whole manifold, rather than just on the open subset.

Q: What are the conditions that must be satisfied in order for an extension of a smooth map to exist?

A: The conditions that must be satisfied in order for an extension of a smooth map to exist include:

• Smoothness: The map must be smooth on the open subset.
• Compactness: The manifolds must be compact.
• Connectedness: The manifolds must be connected.
• Local triviality: The map must be locally trivial, meaning that it can be represented by a smooth function on a neighborhood of each point in the open subset.

Q: What are some methods that can be used to construct an extension of a smooth map?

A: Some methods that can be used to construct an extension of a smooth map include:

• Partition of unity: A partition of unity is a collection of smooth functions that are positive on a neighborhood of each point in the manifold and sum to 1 on the whole manifold. Using a partition of unity, we can construct an extension of the map by multiplying the map by a smooth function that is 1 on the open subset and 0 outside of the open subset.
• Smooth approximation: A smooth approximation of a map is a smooth map that is close to the original map. Using a smooth approximation, we can construct an extension of the map by approximating the map on a neighborhood of each point in the open subset and then extending the approximation to the whole manifold.
• Local trivialization: A local trivialization of a map is a smooth map that represents the map on a neighborhood of each point in the open subset. Using a local trivialization, we can construct an extension of the map by representing the map on a neighborhood of each point in the open subset and then extending the representation to the whole manifold.

Q: Can an extension of a smooth map always be constructed using a partition of unity?

A: No, an extension of a smooth map cannot always be constructed using a partition of unity. A partition of unity can only be used to construct an extension of a smooth map if the map is smooth and the manifolds are compact.

Q: What are some examples of smooth maps that can be extended to the whole manifold?

A: Some examples of smooth maps that can be extended to the whole manifold include:

• Identity map: The identity map on a manifold is a smooth map that maps each point to itself. The identity map can be extended to the whole manifold using a partition of unity.
• Constant map: A constant map on a manifold is a smooth map that maps each point to a fixed point. The constant map can be extended to the whole manifold using a smooth approximation.
• Smooth embedding: A smooth embedding of a manifold into manifold is a smooth map that is injective and has a smooth inverse. A smooth embedding can be extended to the whole manifold using a local trivialization.

Q: What are some common mistakes to avoid when extending a smooth map?

A: Some common mistakes to avoid when extending a smooth map include:

• Not checking the conditions for existence: Before attempting to extend a smooth map, it is essential to check that the conditions for existence are satisfied.
• Using the wrong method: Different methods may be suitable for different types of maps and manifolds. It is essential to choose the correct method for the specific problem.
• Not verifying the smoothness of the extension: After constructing an extension of a smooth map, it is essential to verify that the extension is smooth.

Q: How can I determine if an extension of a smooth map exists?

A: To determine if an extension of a smooth map exists, you can use the following steps:

1. Check the conditions for existence: Verify that the conditions for existence are satisfied, including smoothness, compactness, connectedness, and local triviality.
2. Use a method to construct an extension: Choose a method to construct an extension of the smooth map, such as partition of unity, smooth approximation, or local trivialization.
3. Verify the smoothness of the extension: After constructing an extension of the smooth map, verify that the extension is smooth.

By following these steps, you can determine if an extension of a smooth map exists and construct the extension if it does.