- Thread starter
- #1

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:

**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 \(\displaystyle B \subset H^k\) ... ... ?

Help will be much appreciated ...

Peter