Discussion Overview
The discussion centers around the classification of the unit square, [0,1]^2, as a smooth 2-manifold in \mathbb{R}^2, particularly addressing the implications of its corners and boundaries. Participants explore the definitions and properties of manifolds, smooth manifolds, and submanifolds, as well as the conditions under which the unit square can be considered a manifold with boundary.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants assert that the unit square is a smooth 2-manifold but question the presence of corners, particularly at (1,1).
- Others argue that the unit square is not a smooth submanifold of \mathbb{R}^2 due to the lack of a tangent space at its corners.
- A participant highlights that the unit square can be homeomorphic to a unit disc, which is a manifold with boundary, suggesting it can have a smooth structure despite its corners.
- There is a discussion on the definition of a manifold, with some participants emphasizing the need for transition functions to be smooth, while others question the implications of using a single chart in the atlas.
- Some participants express confusion over the terminology of "submanifold" and its distinction from a manifold in \mathbb{R}^n.
- One participant proposes a specific definition of a manifold that includes conditions on the rank of the derivative of coordinate maps, raising questions about reconciling this with the notion of corners or cusps in smooth manifolds.
Areas of Agreement / Disagreement
Participants do not reach a consensus on whether the unit square can be classified as a smooth manifold in \mathbb{R}^2. There are multiple competing views regarding the definitions and implications of smoothness, corners, and the nature of manifolds with boundaries.
Contextual Notes
Participants note that the definitions of manifolds and smooth manifolds can vary, leading to different interpretations of the unit square's classification. The discussion highlights the importance of transition functions and the conditions required for a structure to be considered smooth.