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Manifolds with Boundary - Shastri Definition 3.2.1 and Remark 3.2.1 ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
I am reading Anant R. Shastri's book: "Elements of Differential Topology" ... ...

I am currently focused on Chapter 3: Submanifolds of Euclidean Spaces ... ... and in particular Section 3.2: Manifolds with Boundary ...

I am having trouble understanding Shastri's Remark following Definition 3.2.1 ...

Definition 3.2.1 and the remark following read as follows:



Shastri - 1 - Defn 3.2.1 & Remark3.2.1 ... PART 1 ... .png
Shastri - 2 - Defn 3.2.1 & Remark3.2.1 ... PART 2 ... .png



General Request

I am somewhat confused by Shastri's argument that

1. If \(\displaystyle x \in \partial X\) ... then for every chart \(\displaystyle \psi\) of \(\displaystyle X\) at \(\displaystyle x\), it follows that \(\displaystyle \psi (x) \in \mathbb{R}^{ k-1 } \times 0\)

and

2. It follows that \(\displaystyle X \text { minus } \partial X = \{ x \in X \ : \ \phi (x) \in \text{ int } ( H^k) \text{ for some chart } \phi \} \)


If anyone can clarify why 1 and 2 follow ... using a better, simpler and/or more detailed argument than Shastri ... then I would be most grateful ...






Question

In Figure 1 below, I have tried to summarize what Shastri argues cannot happen ... that is set \(\displaystyle A\) containing the point \(\displaystyle y\) is not in \(\displaystyle \mathbb{R}^{ k-1 } \times 0\) ... but why exactly can't this happen ... ?

Further ... why exactly is impossible for a neighborhood of \(\displaystyle z\) in \(\displaystyle \mathbb{R}^k\) to be contained in \(\displaystyle B \subset H^k\) ... ... ?



Shastri - Figure 1 pertaining to Remark 3.2.1 ... .png





Help will be much appreciated ...

Peter