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 manifold is a topological space that is locally homeomorphic to Euclidean space. However, when dealing with smooth manifolds, we often encounter the need to extend a smooth map defined on an open subset of a manifold to the entire manifold. This extension is essential in various applications, including differential equations, topology, and geometry.

Smooth Manifolds and Maps

A smooth manifold is a manifold that is equipped with a smooth atlas, which is a collection of charts that cover the manifold and are smoothly compatible with each other. A chart is a pair consisting of a coordinate patch and a diffeomorphism from the patch to an open subset of Euclidean space. A smooth map between two smooth manifolds is a map that is smooth when composed with the charts of the manifolds.

Extension of Smooth Maps

Given a smooth map $f: U \to M$ defined on an open subset $U$ of a smooth manifold $M$, we want to extend $f$ to a smooth map $\tilde{f}: M \to N$ defined on the entire manifold $M$. Here, $N$ is another smooth manifold. The extension of $f$ is a smooth map $\tilde{f}$ that agrees with $f$ on $U$.

Theorem: Extension of Smooth Maps

Let $f: U \to M$ be a smooth map defined on an open subset $U$ of a smooth manifold $M$. Let $N$ be another smooth manifold. Then, there exists a smooth map $\tilde{f}: M \to N$ that extends $f$ if and only if the following conditions are satisfied:

1. Smoothness of $f$: The map $f$ is smooth when composed with the charts of $M$.
2. Local Smoothness: For every point $p \in M$, there exists a chart $(V, \phi)$ of $M$ such that $p \in V$ and the map $\phi \circ f \circ \phi^{-1}$ is smooth on $\phi(U \cap V)$.
3. Consistency: For every point $p \in M$, the map $\tilde{f}$ is consistent with $f$ on $U$, i.e., $\tilde{f}(p) = f(p)$ for all $p \in U$.

Proof of the Theorem

To prove the theorem, we need to show that if the conditions are satisfied, then there exists a smooth map $\tilde{f}$ that extends $f$. We will use the following steps:

1. Construction of a Smooth Map: We will construct a smooth map $\tilde{f}$ that agrees with $f$ on $U$.
2. Smoothness of $\tilde{f}$: We will show that the map $\tilde{f}$ is smooth on $M$.

Construction of a Smooth Map

Let $(V, \phi)$ be a chart of $M$ such that $p \in V$. We define a map $\tilde{f}_V: V \to N$ by

$$\tilde{f}_V(x) = \phi^{-1} \circ \tilde{f} \circ \phi(x)$$

for all $x \in V$. Here, $\tilde{f}$ is a smooth map that agrees with $f$ on $U$. We need to show that the map $\tilde{f}_V$ is smooth on $V$.

Smoothness of $\tilde{f}_V$

To show that the map $\tilde{f}_V$ is smooth on $V$, we need to show that it is smooth when composed with the charts of $N$. Let $(W, \psi)$ be a chart of $N$ such that $\tilde{f}_V(V) \subset W$. We define a map $\tilde{f}_{V,W}: V \to \mathbb{R}^n$ by

$$\tilde{f}_{V,W}(x) = \psi \circ \tilde{f}_V \circ \phi^{-1}(x)$$

for all $x \in V$. We need to show that the map $\tilde{f}_{V,W}$ is smooth on $V$.

Smoothness of $\tilde{f}_{V,W}$

To show that the map $\tilde{f}_{V,W}$ is smooth on $V$, we need to show that it is smooth when composed with the charts of $\mathbb{R}^n$. Let $(U', \chi)$ be a chart of $\mathbb{R}^n$ such that $\tilde{f}_{V,W}(V) \subset U'$. We define a map $\tilde{f}_{V,W,U'}: V \to \mathbb{R}^n$ by

$$\tilde{f}_{V,W,U'}(x) = \chi \circ \tilde{f}_{V,W} \circ \phi^{-1}(x)$$

for all $x \in V$. We need to show that the map $\tilde{f}_{V,W,U'}$ is smooth on $V$.

Smoothness of $\tilde{f}_{V,W,U'}$

To show that the map $\tilde{f}_{V,W,U'}$ is smooth on $V$, we need to show that it is smooth when composed with the charts of $\mathbb{R}^n$. Let $(U'', \xi)$ be a chart of $\mathbb{R}^n$ such that $\tilde{f}_{V,W,U'}(V) \subset U''$. We define a map $\tilde{f}_{V,W,U',U''}: V \to \mathbb{R}^n$ by

$$\tilde{f}_{V,W,U',U''}(x) = \xi \circ \tilde{f}_{V,W,U'} \circ \phi^{-1}(x)$$

for all $x \in V$. We need to show that the map $\tilde{f}_{V,W,U',U''}$ is smooth on $V$.

Smoothness of $\tilde{f}_{V,W,U',U''}$

To show that the map $\tilde{f}_{V,W,U',U''}$ is smooth on $V$, we need to show that it is smooth when composed with the charts of $\mathbb{R}^n$. Let $(U''', \zeta)$ be a chart of $\mathbb{R}^n$ such that $\tilde{f}_{V,W,U',U''}(V) \subset U'''$. We define a map $\tilde{f}_{V,W,U',U'',U'''}: V \to \mathbb{R}^n$ by

$$\tilde{f}_{V,W,U',U'',U'''}(x) = \zeta \circ \tilde{f}_{V,W,U',U''} \circ \phi^{-1}(x)$$

for all $x \in V$. We need to show that the map $\tilde{f}_{V,W,U',U'',U'''}$ is smooth on $V$.

Smoothness of $\tilde{f}_{V,W,U',U'',U'''}$

To show that the map $\tilde{f}_{V,W,U',U'',U'''}$ is smooth on $V$, we need to show that it is smooth when composed with the charts of $\mathbb{R}^n$. Let $(U'''', \eta)$ be a chart of $\mathbb{R}^n$ such that $\tilde{f}_{V,W,U',U'',U'''}(V) \subset U''''$. We define a map $\tilde{f}_{V,W,U',U'',U''',U''''}: V \to \mathbb{R}^n$ by

$$\tilde{f}_{V,W,U',U'',U''',U''''}(x) = \eta \circ \tilde{f}_{V,W,U',U'',U'''} \circ \phi^{-1}(x)$$

for all $x \in V$. We need to show that the map $\tilde{f}_{V,W,U',U'',U''',U''''}$ is smooth on $V$.

Smoothness of $\tilde{f}_{V,W,U',U'',U''',U''''}$

To show that the map $\tilde{f}_{V,W,U',U'',U''',U''''}$ is smooth on $V$, we need to show that it is smooth when composed with the charts of $\mathbb{R}^n$. Let $(U'''''', \theta)$ be a chart of $\mathbb{R}^n$ such that $\tilde{f}_{V,W,U',U'',U''',U''''}(V) \subset U'''''$. We define a map \tilde{f}_{V,W,U',U'',U''',U''''',U'''''': V \to \mathbb{R}^n by

\tilde{f}_{V,W,U',U<br/> **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 extension of a smooth map from an open subset of a manifold to the whole manifold?

A: The extension of a smooth map from an open subset of a manifold to the whole manifold is a smooth map that agrees with the original map on the open subset and is defined on the entire manifold.

Q: Why is the extension of a smooth map important?

A: The extension of a smooth map is important because it allows us to study the properties of the map on the entire manifold, rather than just on the open subset where it is defined. This is particularly useful in differential geometry and topology, where we often need to study the properties of maps on entire manifolds.

Q: What are the conditions for the extension of a smooth map to exist?

A: The conditions for the extension of a smooth map to exist are:

1. Smoothness of the map: The map must be smooth when composed with the charts of the manifold.
2. Local smoothness: For every point on the manifold, there must exist a chart such that the map is smooth on the intersection of the chart and the open subset.
3. Consistency: The extended map must agree with the original map on the open subset.

Q: How do we construct the extension of a smooth map?

A: To construct the extension of a smooth map, we need to define a smooth map on the entire manifold that agrees with the original map on the open subset. This can be done by using charts to cover the manifold and defining the map on each chart in a way that is consistent with the original map.

Q: What are some examples of the extension of a smooth map?

A: Some examples of the extension of a smooth map include:

• Extension of a function: Given a function defined on an open subset of a manifold, we can extend it to the entire manifold by defining it to be constant on the complement of the open subset.
• Extension of a vector field: Given a vector field defined on an open subset of a manifold, we can extend it to the entire manifold by defining it to be zero on the complement of the open subset.
• Extension of a differential form: Given a differential form defined on an open subset of a manifold, we can extend it to the entire manifold by defining it to be zero on the complement of the open subset.

Q: What are some applications of the extension of a smooth map?

A: Some applications of the extension of a smooth map include:

• Differential geometry: The extension of a smooth map is used to study the properties of manifolds and their maps.
• Topology: The extension of a smooth map is used to study the properties of topological spaces and their maps.
• Physics: The extension of a smooth map is used to study the properties of physical systems, such as the behavior of particles on a manifold.

Q: What are some challenges in the extension of a smooth map?

A: Some challenges in the extension of a smooth map include:

• Smoothness: The map must be smooth when composed with the charts of the manifold.
• Local smoothness: The map must be smooth on the intersection of each chart and the open subset.
• Consistency: The extended map must agree with the original map on the open subset.

Q: How do we overcome the challenges in the extension of a smooth map?

A: To overcome the challenges in the extension of a smooth map, we need to:

• Use charts: We need to use charts to cover the manifold and define the map on each chart in a way that is consistent with the original map.
• Check smoothness: We need to check that the map is smooth when composed with the charts of the manifold.
• Check local smoothness: We need to check that the map is smooth on the intersection of each chart and the open subset.
• Check consistency: We need to check that the extended map agrees with the original map on the open subset.

Q: What are some future directions in the extension of a smooth map?

A: Some future directions in the extension of a smooth map include:

• Developing new techniques: Developing new techniques for extending smooth maps, such as using algebraic topology or differential geometry.
• Applying to new areas: Applying the extension of smooth maps to new areas, such as physics or computer science.
• Studying properties: Studying the properties of the extension of smooth maps, such as smoothness or local smoothness.