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Chapter 8: Problem 3

Let \(M\) be a nonempty positive-dimensional smooth manifold with or without boundary. Show that \(\mathfrak{X}(M)\) is infinite-dimensional.

Short Answer

Expert verified
The space of smooth vector fields \( \mathfrak{X}(M) \) is infinite-dimensional because we can construct infinitely many distinct fields by varying smooth functions in any local chart.

Step by step solution

01

Understanding the Definitions

First, recognize that \( \mathfrak{X}(M) \) denotes the space of all smooth vector fields on a manifold \( M \). This includes vector fields that are smooth and tangential at every point on the manifold.
02

Considering Local Coordinates

Since \( M \) is positive-dimensional, we can locally describe \( M \) using coordinates in a chart. Around any point \( p \in M \), there exists a coordinate neighborhood \( (U, \varphi) \), where \( \varphi : U \rightarrow \mathbb{R}^n \) is a diffeomorphism for some \( n > 0 \).
03

Constructing Local Vector Fields

In each coordinate patch \( U \), we can choose smooth vector fields corresponding to each coordinate direction. For example, in a local chart, we can define vector fields \( \frac{\partial}{\partial x^i} \) for each local coordinate \( x^i \), where \( i = 1, 2, \, \ldots, \, n \).
04

Explanation of Infinite-dimensionality

Since \( M \) is positive-dimensional, there are infinitely many smooth functions on \( U \). For each smooth function \( f : U \rightarrow \mathbb{R} \), we can form a smooth vector field \( f \frac{\partial}{\partial x^i} \) that is distinct from other vector fields formed with different \( f \)'s or \( i \)'s.
05

Conclusion on Infinite-dimensionality

As we can form infinitely many distinct smooth vector fields by varying the smooth function \( f \), the space \( \mathfrak{X}(M) \) is spanned by an infinite-dimensional vector space, consisting of such local vector fields patched together covering \( M \).

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Key Concepts

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

Smooth Manifolds
A smooth manifold is a foundational concept in mathematics, especially in geometry and topology. Imagine it as a shape or surface that at each point resembles the flat space we are familiar with, like a plane or higher-dimensional Euclidean space. What makes a manifold 'smooth' is that it can be nicely described using smooth functions, which means no sudden breaks or corners.

Key characteristics of smooth manifolds include:
  • **Local Resemblance**: Locally, around each point, a manifold looks like a simple space, such as a straight line or plane.
  • **Charts and Atlases**: A manifold is often covered by a collection of charts, which are maps from parts of the manifold to Euclidean space, ensuring smooth transitions between overlapping parts.
  • **Dimensionality**: The dimension of a manifold is constant, indicating the number of coordinates needed to describe a point on it.
Manifolds can exist in any dimension, and their beauty lies in their flexibility and smoothness. From surfaces like spheres to complex shapes used in advanced physics, manifolds form the language we use to describe space and form.
Vector Fields
Vector fields are an essential topic when studying manifolds. You can think of them as assigning a vector, which has both direction and magnitude, to every point on a manifold. This helps in understanding the manifold's geometry and dynamics.

In detail, vector fields on a manifold:
  • **Assign Vectors Continuously**: A vector field smoothly associates vectors to points, ensuring there are no jumps or discontinuities.
  • **Tangent Vectors**: At every point, the vector lies in the tangent space of the manifold, giving us a "directional" property.
  • **Applications**: Vector fields can describe many physical concepts, like wind patterns on Earth or the flow of a liquid.
In mathematics, especially differential geometry, vector fields are represented systematically and can be combined with functions to analyze and explore new properties leading to advanced study topics.
Smooth Functions
Smooth functions are the building blocks for exploring manifolds and performing complex mathematical analysis. A function is called smooth if it is differentiable and has continuous derivatives up to any order, which flexible allows for exploration and computation.

Important aspects of smooth functions are:
  • **Differentiability**: A smooth function can be derived multiple times, with each derivative being continuous.
  • **Role in Manifolds**: They are used to define smooth charts and establish connections between different parts of a manifold.
  • **Construction of Vector Fields**: Smooth functions often combine with vector fields to form new vector fields, crucial in exploring the structure and behaviors on manifolds.
Smooth functions offer the basis for developing and solving equations on manifolds and help maintain the overall continuity and differentiability of structures in mathematical models.
Coordinate Charts
Coordinate charts are a pivotal part of how we understand and work with manifolds. They serve as local maps that translate points on a manifold to Euclidean spaces, enabling us to apply various mathematical tools.

Key features of coordinate charts include:
  • **Local Addressing**: Ordinarily, a manifold doesn't have global coordinates, but charts facilitate a local coordinate system around a point, easing computation.
  • **Diffeomorphisms**: The transition functions between overlapping charts are smooth and reversible, which maintains the integrity and smoothness of the manifold.
  • **Atlas**: An atlas is a collection of overlapping charts that fully cover the manifold, ensuring comprehensive mapping and exploration.
Coordinate charts help in breaking down complex problems on manifolds into manageable local problems, which can be tackled using familiar mathematical techniques from calculus and algebra.

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Most popular questions from this chapter

For each of the following vector fields on the plane, compute its coordinate representation in polar coordinates on the right half-plane \(\\{(x, y): x>0\\}\). (a) \(X=x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}\). (b) \(Y=x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\). (c) \(Z=\left(x^{2}+y^{2}\right) \frac{\partial}{\partial x}\).

Let \(A\) be any algebra over \(\mathbb{R}\). A derivation of \(A\) is a linear map \(D: A \rightarrow A\) satisfying \(D(x y)=(D x) y+x(D y)\) for all \(x, y \in A\). Show that if \(D_{1}\) and \(D_{2}\) are derivations of \(A\), then \(\left[D_{1}, D_{2}\right]=D_{1} \circ D_{2}-D_{2} \circ D_{1}\) is also a derivation. Show that the set of derivations of \(A\) is a Lie algebra with this bracket operation.

(a) Given Lie algebras \(g\) and \(h\), show that the direct sum \(g \oplus h\) is a Lie algebra with the bracket defined by $$ \left[(X, Y),\left(X^{\prime}, Y^{\prime}\right)\right]=\left(\left[X, X^{\prime}\right],\left[Y, Y^{\prime}\right]\right) \text {. } $$ (b) Suppose \(G\) and \(H\) are Lie groups. Prove that \(\operatorname{Lie}(G \times H)\) is isomorphic to \(\operatorname{Lie}(G) \oplus \operatorname{Lie}(H)\).

Let \(A \subseteq \mathfrak{X}\left(\mathbb{R}^{3}\right)\) be the subspace spanned by \(\\{X, Y, Z\\}\), where $$ X=y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}, \quad Y=z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}, \quad Z=x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x} $$ Show that \(A\) is a Lie subalgebra of \(\mathfrak{X}\left(\mathbb{R}^{3}\right)\), which is isomorphic to \(\mathbb{R}^{3}\) with the cross product. (Used on p. 538.)

Suppose \(F: M \rightarrow N\) is a smooth submersion, where \(M\) and \(N\) are positivedimensional smooth manifolds. Given \(X \in \mathfrak{X}(M)\) and \(Y \in \mathfrak{X}(N)\), we say that \(X\) is a lift of \(Y\) if \(X\) and \(Y\) are \(F\)-related. A vector field \(V \in \mathfrak{X}(M)\) is said to be vertical if \(V\) is everywhere tangent to the fibers of \(F\) (or, equivalently, if \(V\) is \(F\)-related to the zero vector field on \(N\) ). (a) Show that if \(\operatorname{dim} M=\operatorname{dim} N\), then every smooth vector field on \(N\) has a unique lift. (b) Show that if \(\operatorname{dim} M \neq \operatorname{dim} N\), then every smooth vector field on \(N\) has a lift, but that it is not unique. (c) Assume in addition that \(F\) is surjective. Given \(X \in \mathfrak{X}(M)\), show that \(X\) is a lift of a smooth vector field on \(N\) if and only if \(d F_{p}\left(X_{p}\right)=d F_{q}\left(X_{q}\right)\) whenever \(F(p)=F(q)\). Show that if this is the case, then \(X\) is a lift of a unique smooth vector field. (d) Assume in addition that \(F\) is surjective with connected fibers. Show that a vector field \(X \in \mathfrak{X}(M)\) is a lift of a smooth vector field on \(N\) if and only if \([V, X]\) is vertical whenever \(V \in \mathscr{X}(M)\) is vertical.
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