Problem 5 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Let \(\mathbb{R}^{\prime}\) be the marifold consisting of \(\mathbb{R}\) with differentiable structure generated by the chart \(\left(R ; y=x^{3}\right)\). Show that the identity map id \(\mathbb{R}: \mathbb{R}^{\prime} \rightarrow \mathbb{R}\) is a differentiable homeomorphism, which is not a diffeomorphism.

Step-by-Step Solution

Answer

The identity map id :\(\mathbb{R}^{\prime}\rightarrow \mathbb{R}\) is a differentiable homeomorphism, because both the identity map and its inverse are continuous and the identity map is differentiable. However, it is not a diffeomorphism because the inverse function, or the transition map from \(\mathbb{R}\) to \(\mathbb{R}^{\prime}\), is not differentiable at 0.

Step 1: Check if id :\(\mathbb{R}^{\prime}\rightarrow \mathbb{R}\) is continuous

The identity map is always continuous, because it is defined as id:\(\mathbb{R}^{\prime}\rightarrow \mathbb{R}\), \(x\rightarrow x\). For the continuity of a real-valued function id\(x\), limit of id\(x_{n}\) as \(n\) approaches infinity is equal to id\(x\), for every sequence \(x_{n}\) converging to \(x\). Therefore id :\(\mathbb{R}^{\prime}\rightarrow \mathbb{R}\) is continuous.

Step 2: Check if the inverse map id :\(\mathbb{R}\rightarrow \mathbb{R}^{\prime}\) is continuous

The inverse map is also the identity, because it is defined as id:\(\mathbb{R}\rightarrow \(\mathbb{R}^{\prime}\), \(x\rightarrow x\). And using the similar reasoning as in Step 1, id :\(\mathbb{R}\rightarrow \mathbb{R}^{\prime}\) is continuous.

Step 3: Check the differentiability of id :\(\mathbb{R}^{\prime}\rightarrow \mathbb{R}\)

Since \(\mathbb{R}^{\prime}\) has the differentiable structure generated by the chart \(\left(R ; y=x^{3}\right)\), we need to check differentiability in this chart. The derivative of the identity function on \(\mathbb{R}\) in the chart is always 1. Hence, the map id :\(\mathbb{R}^{\prime}\rightarrow \mathbb{R}\) is differentiable.

Step 4: Check if id :\(\mathbb{R}\rightarrow \mathbb{R}^{\prime}\) is differentiable

To be a diffeomorphism, the inverse function also needs to be differentiable. Transition map from \(\mathbb{R}\) to \(\mathbb{R}^{\prime}\) is \(x^{3}\), and its derivative at 0 does not exist. Hence, id :\(\mathbb{R}\rightarrow \mathbb{R}^{\prime}\) is not differentiable.

Key Concepts

Manifold

In the world of differential geometry, a **manifold** is a space that resembles regular Euclidean space (\(\mathbb{R}^n\)) around each point, at least on a small scale. Think of it as a surface that can be stretched or bent, like the surface of a sphere or a torus. Manifolds are a generalization of curves and surfaces.
What makes manifolds fascinating is that:

• They allow us to work with shapes that are not flat, but can still be analyzed with calculus.
• Each tiny patch on the manifold is "locally" like \(\mathbb{R}^n\).
• Manifolds can have different dimensions, like 1D (line), 2D (surface), and beyond.

The manifold mentioned in the exercise, \(\mathbb{R}'\), uses the same points as \(\mathbb{R}\) but applies a different structure due to a differentiable chart, making it interesting to study how smooth maps interact in this space.

Differentiable Structure

A **differentiable structure** on a manifold provides the ability to do calculus on the manifold. It is defined through charts, which are maps that cover the manifold and relate it to \(\mathbb{R}^n\). When these charts overlap, they must smoothly transition from one to another.
For this exercise:

• The chart \((R ; y = x^3)\) defines how \(\mathbb{R}'\) differs from the usual \(\mathbb{R}\).
• This chart "transforms" it by mapping \(x\) to \(y = x^3\), introducing a special way of considering differentiation.
• This transformation affects how functions like the identity map behave in terms of smoothness.

Differentiable structures allow us to explore differences in smoothness between different manifolds, even if they share the same point set.

Diffeomorphism

A **diffeomorphism** is a special type of map between manifolds. It is both a bijection and "infinitely" smooth, meaning that both the map and its inverse are differentiable.
Key points about diffeomorphisms:

• They preserve the differentiable structure, making them important in understanding manifold equivalence.
• In the exercise, the identity map from \(\mathbb{R}'\) to \(\mathbb{R}\) is smooth, but its inverse, due to the \(x^3\) structure, is not. Thus, it is not a diffeomorphism.
• This differentiability requirement is strict, revealing subtle nuances in structure between \(\mathbb{R}\) and \(\mathbb{R}'\).

Understanding diffeomorphisms helps in identifying when two manifold structures are genuinely similar or fundamentally different.

Homeomorphism

A **homeomorphism** is a different kind of map that focuses on the topological, rather than smooth, properties of spaces. It requires that the map be continuous, have a continuous inverse, and be a bijection.
Important aspects of homeomorphisms:

• They ensure that spaces are "equivalent" in a topological sense, without concern for smoothness.
• For the exercise, the identity map is a homeomorphism because it is continuous in both directions, preserving the basic topology of \(\mathbb{R}\).
• Unlike diffeomorphisms, homeomorphisms don't require differentiability.

By understanding homeomorphisms, we see how topology provides a more flexible framework for identifying when spaces can be considered the same, even if they do not allow the same smoothness considerations.