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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,881
Hobart, Tasmania
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 ...

Example 3.1.13 reads as follows:



Lovett- Example 3.1.13 ... .png



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



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



Lovett - 1 - Defn 3.1.11 ... PART 1 ... .png
Lovett - 2 - Defn 3.1.11 ... PART 2 ... .png


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



Lovett - 1 - Defn 3.1.3 ... PART 1 ... .png
Lovett - 2 - Defn 3.1.3 ... PART 2 ... .png





Hope that helps ...

Peter