Homework 1 Problem 5

Problem

Give an example of a map $f: \mathbb{T}^3 \to \mathbb{T}^3$ which is injective but not surjective. Give an example of a map $g: \mathbb{T}^3 \to \mathbb{T}^3$ which is surjective but not injective. Give an example of a map $h: \mathbb{T}^3 \to \mathbb{T}^3$ which is neither surjective nor injective.

Solution

The torus $\mathbb{T}^3 = \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ as seen in Problem 4. The coordinate for $\mathbb{S}^1$ is in the interval $[0,2\pi)$. Let's say the coordinates for $\mathbb{T}^3$ are $\theta_1$, $\theta_2$, and $\theta_3$, each in the interval $[0,2\pi)$. Then, the following is an example of a map $f$ which is injective but not surjective

(1)
\begin{align} \theta_1 &=& .5 \theta_1, \theta_2 &=& \theta_2, \theta_3 &=& \theta_3 \end{align}

Similarly, the following is an example of a map $g$ which is surjective but not injective

(2)
\begin{align} \theta_1 &=& 2 \theta_1, \theta_2 &=& \theta_2, \theta_3 &=& \theta_3 \end{align}

Combining these results yields an example of a map $h$ which is neither injective nor surjective

(3)
\begin{align} \theta_1 &=& 2 \theta_1, \theta_2 &=& .5 \theta_2, \theta_3 &=& \theta_3 \end{align}

Scott agrees!

Discussion

Right?

It's interesting that Marsden and Ratiu define the notion of a coordinate chart $(U, \phi)$ on a manifold $M$ without requiring $\phi(U)$ to be an open subset of $\mathbb{R}^n$. This requirement is sometimes included in the definition, as it is in (for example) Peter Olver's Applications of Lie Groups to Differential Equations.

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