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Chapter 5: Problem 18

Suppose \(M\) is a smooth manifold and \(S \subseteq M\) is a smooth submanifold. (a) Show that \(S\) is embedded if and only if every \(f \in C^{\infty}(S)\) has a smooth extension to a neighborhood of \(S\) in \(M\). [Hint: if \(S\) is not embedded, let \(p \in S\) be a point that is not in the domain of any slice chart. Let \(U\) be a neighborhood of \(p\) in \(S\) that is embedded, and consider a function \(f \in C^{\infty}(S)\) that is supported in \(U\) and equal to 1 at \(p\).] (b) Show that \(S\) is properly embedded if and only if every \(f \in C^{\infty}(S)\) has a smooth extension to all of \(M\).

Short Answer

Expert verified
(a) S is embedded if functions extend to a neighborhood in M. (b) S is properly embedded if functions extend to all of M.

Step by step solution

01

Understanding Embedded Submanifold

To prove that a submanifold \( S \) of \( M \) is embedded, consider that when \( S \) is embedded into \( M \), it inherits the topological properties of \( M \), allowing smooth functions on \( S \) to have smooth extensions to a neighborhood in \( M \). This means the inclusion map \( i: S \rightarrow M \) is a topological embedding.
02

Proving "If" Direction

Assume every smooth function \( f \in C^{\infty}(S) \) has a smooth extension to a neighborhood of \( S \) in \( M \). By taking any point \( p \in S \), there exists a neighborhood \( U \) of \( p \) in \( S \), where \( f \) is defined and can be smoothly extended to a neighborhood in \( M \). Hence \( S \) is embedded trivially.
03

Proving "Only If" Direction

Given \( S \) is an embedded submanifold, for any \( f \in C^{\infty}(S) \), consider that by embeddedness, \( S \) being locally closed means every \( p \in S \) is in a coordinate chart centered at \( p \), allowing \( f \) to extend smoothly to a neighborhood. Thus any function on \( S \) has a smooth extension to \( M \).
04

Using the Hint for Counterexample

If \( S \) is not embedded, then for point \( p \) in \( S \) not lying within a slice of a slice chart, a smooth function \( f \) that is 1 at \( p \) but has no well-defined extension challenges the hypothesis. This demonstrates why no smooth extension could exist for \( S \) not embedded.
05

Understanding Properly Embedded Submanifold

A submanifold \( S \) is properly embedded if it is closed in \( M \) and the inclusion map is a proper map. This means every continuous map from \( S \) which is smooth has a well-defined limit requiring the function to smoothly extend over all of \( M \).
06

Proving Properly Embedded Condition

For \( S \) being properly embedded, assume every function \( f \in C^{\infty}(S) \) can be extended to \( M \). Since every point of \( S \) respects compactness and inclusion is closed, the conditions satisfy a proper embedding and encapsulate \( S \) fully within \( M \), ensuring a full extension grid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Embedded Submanifold
An **embedded submanifold** is a type of smooth manifold that sits nicely within another manifold, inheriting many of its properties. For a submanifold \( S \) to be embedded in a manifold \( M \), every smooth function defined on \( S \) must have the capability to extend smoothly to a neighborhood of \( S \) in \( M \).

This smooth extension is facilitated by the inclusion map, which is continuous and provides a topological embedding of \( S \) into \( M \). This means \( S \) retains the manifold structure and has an open set embedding in \( M \).

**Breaking this down:**
  • Every point \( p \) on \( S \) finds itself in a local slice of the bigger manifold \( M \).
  • All functions \( f: S \to \mathbb{R} \) can "grow" into \( M \) without losing smoothness.
  • If \( S \) cannot precisely embed inside \( M \), a function at \( p \) in \( S \) may fail to extend smoothly.
Understanding these traits is crucial in demonstrating when \( S \) is indeed an embedded submanifold.
Smooth Extension
A **smooth extension** refers to the process by which a smooth function on a submanifold \( S \) is continued to a neighborhood in a manifold \( M \) without losing its smooth properties. It is a vital concept when determining the embeddedness of submanifolds.

To establish a smooth extension:
  • Every smooth function \( f \) on \( S \) must extend beyond \( S \) into nearby sections of \( M \).
  • If \( f \) extends to an open neighborhood, it confirms \( S \)'s embedment.
  • A lack of extension capability often implies \( S \) isn't appropriately embedded.
Imagine each smooth function on \( S \) as a sprout growing roots; these roots need space and soil (neighborhood in \( M \)) to expand smoothly. When \( S \) is equipped to provide that, smooth extension works seamlessly.
Properly Embedded Submanifold
A **properly embedded submanifold** involves more stringent conditions compared to just being embedded. For a submanifold \( S \) to be considered properly embedded in \( M \), every smooth function from \( S \) should smoothly extend over the entire manifold \( M \). This is an upgrade from merely extending into a neighborhood.

**Key Characteristics:**
  • \( S \) must be closed within \( M \), ensuring no edge points are left dangling.
  • The inclusion map from \( S \) to \( M \) needs to be a proper map, meaning pre-images of compact sets in \( M \) are compact in \( S \).
  • Every function on \( S \) seamlessly integrates with \( M \), obliterating any demarcation of boundaries.
This level of embedding is like ensuring \( S \) is fully capable of blending into and becoming one with \( M \) holistically. When \( S \) fulfills these conditions, it suggests a completeness and thorough integration into the larger manifold.

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Most popular questions from this chapter

Let \(F: \mathbb{R}^{2} \rightarrow \mathbb{R}\) be defined by \(F(x, y)=x^{3}+x y+y^{3}\). Which level sets of \(F\) are embedded submanifolds of \(\mathbb{R}^{2}\) ? For each level set, prove either that it is or that it is not an embedded submanifold.

Show by example that an immersed submanifold \(S \subseteq M\) might have more than one topology and smooth structure with respect to which it is an immersed submanifold.
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