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Chapter 3: Problem 2

(a) Suppose \((x, U)\) and \((y, V)\) are two coordinate systems, giving rise to two maps on \(T M\), $$ \begin{aligned} &t_{x}: \pi^{-1}(U) \rightarrow U \times \mathbb{R}^{n}, \quad[x, v]_{q} \mapsto(q, v) \\ &t_{y}: \pi^{-1}(V) \rightarrow V \times \mathbb{R}^{n}, \quad[y, w]_{q} \mapsto(q, w) \end{aligned} $$ Show that in \(\pi^{-1}(U \cap V)\) the sets of the form \(t_{x}^{-1}(A)\) for \(A \subset U \times \mathbb{R}^{n}\) open are exactly the sets of the form \(t_{y}^{-1}(B)\) for \(B \subset V \times \mathbb{R}^{n}\) open. (b) Show that if there is a metric on \(T M\) such that \(t_{x_{i}}\) is a homeomorphism for a collection \(\left(x_{i}, U_{i}\right)\) with \(M=\bigcup_{i} U_{i}\), then all \(t_{x}\) are homeomorphisms. (c) Conclude from Problem 1 that there is a metric on \(T M\) which makes each \(t_{x}\) a homeomorphism.

Short Answer

Expert verified
Preimages of open sets under both mappings coincide. Assuming specific metrics, all mappings become homeomorphisms; therefore for entire manifold T M each mapping shows consistent homeomorphic properties.

Step by step solution

01

Title - Understanding the Problem

Analyze the given coordinate systems \( (x, U) \) and \( (y, V) \), and their mappings \( t_x \) and \( t_y \). We want to show that the preimages of open sets under \( t_x \) and \( t_y \) are the same in \( \pi^{-1}(U \cap V) \).
02

Title - Showing the Preimage Equality

Consider an open set \( A \subset U \times \mathbb{R}^n \). We need to show \( t_x^{-1}(A) \) has a set form in \( t_y \). This implies the composition of the coordinate changes functions in \( \pi^{-1}(U \cap V) \) results in the transformations coinciding in each basis.\r\rFor every point \( p \in \pi^{-1}(U \cap V) \), the changing coordinates will transform to (q, value) that meet same condition \( t_x^{-1}(A) = t_y^{-1}(B) \). Hence, proving preimages are open.
03

Title - Assumption of a Metric and Homeomorphism

Assume some metric allows \( t_{x_i} \) to be homeomorphisms for distinct \( (x_i, U_i) \). Thus, \( [x_i, v_i]_q \rightarrow (q, v_i) \) uniquely. Combined with step 2, this means all \( t_x \) will carry similar properties to their open sets of the manifold.
04

Title - Conclusion from Problem 1

As per Problem 1's precondition, if we align our approach with inserting a metric making \( t_x \) homeomorphism possible, the manifold \( T M \) aligns under identical local mappings. Conclusively, demonstrating all \( t_x \) functions are homeomorphisms.

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Key Concepts

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

Coordinate Systems
Coordinate systems are like a map that helps us navigate through a manifold. Imagine them as local frames that describe positions in different parts of the manifold. For example, we have two coordinate systems \((x, U)\) and \((y, V)\), which create respective mappings \(t_x\) and \(t_y\). These mappings help translate the points and vectors in the manifold to more familiar Euclidean spaces. This makes it easier to perform various computations and visualizations.

In this exercise, we examine how the coordinate systems interact, ensuring that different local descriptions (using \(x\) or \(y\) coordinates) yield consistent information across the overlapping regions \(U \cap V\). This consistency is crucial for our understanding of the structure of the manifold.
Homeomorphisms
A homeomorphism is a kind of transformation that is very special in mathematics. It means that you can stretch, twist, shrink, or bend objects without cutting or gluing them. Mathematically, this means preserving the 'topology' of a space.

In the context of this exercise, we need to show that our coordinate mappings \(t_x\) and \(t_y\) are homeomorphisms. This demonstrates that despite switching between different coordinate views (like maps on a GPS), the underlying structure of the manifold remains intact.

The condition to be a homeomorphism involves being continuous and having a continuous inverse. Essentially, if you can uniquely and smoothly navigate back and forth between two spaces (our original manifold and its coordinate representation), we call it a homeomorphism.
Vector Bundles
Vector bundles add another layer of structure to manifolds. Think of them as assigning a vector space (like a plane with arrows in different directions) to every point on the manifold. It's like having a small 'copy' of a familiar space at each point of a more complex space.

In \(T M\), which stands for the tangent bundle of the manifold \(M\), we associate each point with its tangent space, a space of vectors that 'touch' the manifold at that point. Our mappings \(t_x\) and \(t_y\) help navigate within these vector bundles, translating between local vector spaces and the manifold points.
Preimage Sets
Preimage sets help us understand how different regions or sets in one space transform under a mapping. When we talk about the preimage set \(t_x^{-1}(A)\), we refer to all the points in the original space that map into set \(A\) under the transformation \(t_x\).

In this exercise, we explore the connection between the preimage sets under different coordinate mappings. Specifically, we need to show that the preimage sets under \(t_x\) and \(t_y\) are equivalent in the overlapping region \( \pi^{-1}(U \cap V)\). If they are, it means both coordinate systems describe the same parts of the manifold consistently, even though they may look different locally.
Manifolds
A manifold is a space that looks simple and Euclidean locally but can have a very complex and interesting overall structure. Think of the surface of the Earth; locally, it looks flat (like a piece of paper), but globally, it's a sphere.

In differential geometry, we study these manifolds to understand their properties, using tools like coordinate systems, homeomorphisms, and vector bundles. A key part of this exercise is to ensure that pieces of the manifold (described by different coordinate systems) fit together smoothly, preserving the manifold's structure.

By ensuring that our local coordinate mappings are homeomorphisms, we confirm that the manifold's structure remains intact, regardless of how we locally 'picture' or navigate through it with different coordinates.

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Most popular questions from this chapter

(a) Let \(p_{0} \in S^{n-1}\) be the point \((0, \ldots, 0,1)\). For \(n \geq 2\) define \(f: \mathrm{SO}(n) \rightarrow\) \(S^{n-1}\) by \(f(A)=A\left(p_{0}\right)\). Show that \(f\) is continuous and open. Show that \(f^{-1}\left(p_{0}\right)\) is homeomorphic to \(\mathrm{SO}(n-1)\), and then show that \(f^{-1}(p)\) is homeomorphic to \(\mathrm{SO}(n-1)\) for all \(p \in S^{n-1}\). (b) \(\mathrm{SO}(\mathrm{l})\) is a point, so it is connected. Using part (a), and induction on \(n\), prove that \(\mathrm{SO}(n)\) is connected for all \(n \geq 1\). (c) Show that \(\mathrm{O}(n)\) has exactly two components.

If \(g: \mathbb{R} \rightarrow \mathbb{R}\) is \(C^{\infty}\) show that $$ g(x)=g(0)+g^{\prime}(0) x+x^{2} h(x) $$ for some \(C^{\infty}\) function \(h: \mathbb{R} \rightarrow \mathbb{R}\)

(a) If \(M\) and \(N\) are \(C^{\infty}\) manifolds, and \(\pi_{M}\) [or \(\left.\pi_{N}\right]: M \times N \rightarrow M\) [or \(\left.N\right]\) is the projection on \(M\) [or \(N]\), then \(T(M \times N) \simeq \pi_{M}^{*}(T M) \oplus \pi_{N}^{*}(T N)\). (b) If \(M\) and \(N\) are orientable, then \(M \times N\) is orientable. (c) If \(M \times N\) is orientable, then both \(M\) and \(N\) are orientable.

Let \(M^{n} \subset \mathbb{R}^{N}\) be a \(C^{\infty} n\)-dimensional submanifold. By a chord of \(M\) we mean a point of \(\mathbb{R}^{N}\) of the form \(p-q\) for \(p, q \in M\). (a) Prove that if \(N>2 n+1\), then there is a vector \(v \in S^{N-1}\) such that (i) no chord of \(M\) is parallel to \(v\), (ii) no tangent plane \(M_{p}\) contains \(v\).(b) Let \(\mathbb{R}^{N-1} \subset \mathbb{R}^{N}\) be the subspace perpendicular to \(v\), and \(\pi: \mathbb{R}^{N} \rightarrow \mathbb{R}^{N-1}\) the corresponding projection. Show that \(\pi \mid M\) is a one-one immersion. In particular, if \(M\) is compact, then \(\pi \mid M\) is an imbedding. (c) Every compact \(C^{\infty} n\)-dimensional manifold can be imbedded in \(\mathbb{R}^{2 n+1}\). Note: This is the easy case of Whitney's classical theorem, which gives the same result even for non-compact manifolds (H. Whitney, Differentiable manifolds, Ann. of Math. \(37(1935), 645-680)\). Proofs may be found in Auslander and MacKenzie, Introduction to Differentiable Manifolds and Sternberg, Lectures on Differential Geometry. In Munkres, Elementary Differential Topology, there is a different sort of argument to prove that a not-necessarily-compact \(n\)-manifold \(M\) can be imbedded in some \(\mathbb{R}^{N}\) (in fact, with \(\left.N=(n+1)^{2}\right) .\) Then we may show that \(M\) imbeds in \(\mathbb{R}^{2 n+1}\) using essentially the argument above, together with the existence of a proper map \(f: M \rightarrow \mathbb{R}\), given by Problem '2-30 (compare Guillemin and Pollack, Differential Topology). A much harder result of Whitney shows that \(M^{n}\) can actually be imbedded in \(\mathbb{R}^{2 n}\) (H. Whitney, The self-intersections of a smooth n-manifold in \(2 n\)-space, An?. of Math. \(45(1944), 220-246) .\) Hint: Consider certain naps from appropriate open subsets of \(M \times M\) and \(T M\) to \(S^{N-1}\)

(a) Given an \(n\)-plane bundle \(\xi=\pi: E \rightarrow B\) and an \(m\)-plane bundle \(\eta=\) \(\pi^{\prime}: E^{\prime} \rightarrow B\), let \(E^{\prime \prime} \subset E \times E^{\prime}\) be the set of all pairs \(\left(e, e^{\prime}\right)\) with \(\pi(e)=\pi^{\prime}\left(e^{\prime}\right)\). Let \(\pi^{\prime \prime}\left(e, e^{\prime}\right)=\pi(e)=\pi^{\prime}\left(e^{\prime}\right)\). Show that \(\pi^{\prime \prime}: E^{\prime \prime} \rightarrow B\) is an \((n+m)\)-plane bundle. It is called the Whitney sum \(\xi \oplus \eta\) of \(\xi\) and \(\eta\); the fibre of \(\xi \oplus \eta\) over \(p\) is the direct sum \(\pi^{-1}(p) \oplus \pi^{\prime-1}(p)\). (b) If \(f: Y \rightarrow B\), show that \(f^{*}(\xi \oplus \eta) \simeq f^{*}(\xi) \oplus f^{*}(\eta)\).(c) Given bundles \(\xi_{i}=\pi_{i}: E_{i} \rightarrow B_{i}\), define \(\pi: E_{1} \times E_{2} \rightarrow B_{1} \times B_{2}\) by \(\pi\left(e_{1}, e_{2}\right)=\left(\pi_{1}\left(e_{1}\right), \pi_{2}\left(e_{2}\right)\right)\). Show that this is a bundle \(\xi_{1} \times \xi_{2}\) over \(B_{1} \times B_{2}\) (d) If \(\Delta: B \rightarrow B \times B\) is the "diagonal map", \(\Delta(x)=(x, x)\), show that \(\xi \oplus \eta \simeq\) \(\Delta^{*}(\xi \times \eta)\) (e) If \(\xi\) and \(\eta\) are orientable, show that \(\xi \oplus \eta\) is orientable. (f) If \(\xi\) is orientable, and \(\eta\) is non-orientable, show that \(\xi \oplus \eta\) is also non orientable. (g) Define a "natural" orientation on \(V \oplus V\) for any vector space \(V\), and use this to show that \(\xi \oplus \xi\) is always orientable. (h) If \(X\) is a "figure eight" (c.f. Problem 5), find two non-orientable 1 -plane bundles \(\xi\) and \(\eta\) over \(X\) such that \(\xi \oplus \eta\) is also non-orientable.
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