# Manifolds with Boundary - Lovett, Example 3.1.13 ... ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Stephen Lovett's book: "Differential Geometry of Manifolds" ...

Currently I am focused on Chapter 3: Differential Manifolds ...

I need help in order to fully understand Example 3.1.13 ...

My questions are as follows:

Question 1

In the above Example from Lovett we read the following:

" ... ... The image of $$\displaystyle \vec{X}$$ is a half-torus $$\displaystyle M$$ with $$\displaystyle y \ge 0$$, which to conform to Definition 3.1.11, is easily covered by four coordinate patches ... ... "

Can someone please explain and demonstrate how $$\displaystyle M$$ can be covered by four coordinate patches ... ...

Question 2

Can someone please explain/demonstrate why/how the boundary $$\displaystyle \partial M$$ is the disconnected manifold with $$\displaystyle R_+$$ and $$\displaystyle R_{ - }$$as the two connected components ... ...

Help will be much appreciated ... ...

Peter

========================================================================================

The above post refers to Lovett Definition 3.1.11 ... so I am providing access to the same ... as follows:

Since Definition 3.1.11 above refers to Definition 3.1.3 I am providing access to the same ... as follows:

Hope that helps ...

Peter