tag:blogger.com,1999:blog-4005750306801645234.post4190265318031590584..comments2024-04-20T13:45:15.715+02:00Comments on Spacetime and Geometry: Exercise 2.01 One chart for infinite cylinderUnknownnoreply@blogger.comBlogger2125tag:blogger.com,1999:blog-4005750306801645234.post-27414107925775774422024-02-02T20:17:24.676+01:002024-02-02T20:17:24.676+01:00Preliminary Concepts
**Manifold:** An \(n\)-dimen...Preliminary Concepts<br /><br />**Manifold:** An \(n\)-dimensional manifold is a topological space that locally resembles Euclidean space \( \mathbb{R}^n \), but globally can have a more complex topology.<br /><br />**Chart and Atlas:** A chart on a manifold is a homeomorphism from an open subset of the manifold to an open subset of \( \mathbb{R}^n \). An atlas is a collection of charts that cover the entire manifold. If a manifold can be covered by a single chart, this chart provides global coordinates for the manifold.<br /><br />The Infinite Cylinder \( \mathbb{R} \times S^1 \)<br /><br />The infinite cylinder can be conceptualized as the Cartesian product of the real line \( \mathbb{R} \) and the circle \( S^1 \). The circle can be represented as \( S^1 = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \} \) or, alternatively, through the identification \( S^1 \cong \mathbb{R} / \mathbb{Z} \), where each point on \( S^1 \) corresponds to an equivalence class of real numbers under the equivalence relation \( x \sim y \) if \( x - y \) is an integer.<br /><br />Explicit Map Construction<br /><br />To cover \( \mathbb{R} \times S^1 \) with a single chart, we need to provide a homeomorphism from this manifold to an open space in \( \mathbb{R}^2 \). A natural way to do this is using angular and radial coordinates, but since the "radius" in our cylinder is constant (for \( S^1 \)), we only need to consider the "height" along \( \mathbb{R} \) and the angle in \( S^1 \).<br /><br />We define the function \( \phi: \mathbb{R} \times S^1 \rightarrow \mathbb{R}^2 \) by:<br />\[ \phi(t, \theta) = (t, \theta) \]<br />where \( t \) is the coordinate along \( \mathbb{R} \), representing the height on the cylinder, and \( \theta \) is the angular coordinate on \( S^1 \), identifying \( S^1 \) with the interval \( [0, 2\pi) \) under the identification of \( 0 \) and \( 2\pi \) as the same point.<br /><br />This function is clearly a homeomorphism, as it is bijective, continuous, and its inverse is also continuous. The continuity and bijectivity come from the direct structure of the Cartesian product and the definition of \( S^1 \) and \( \mathbb{R} \). The inverse \( \phi^{-1}: \mathbb{R}^2 \rightarrow \mathbb{R} \times S^1 \) simply takes a point in \( \mathbb{R}^2 \) and assigns it to its corresponding point on the infinite cylinder.<br /><br /> Conclusion<br /><br />We have shown that the infinite cylinder \( \mathbb{R} \times S^1 \) can be covered by a single chart by explicitly constructing a map \( \phi \) that associates each point on the cylinder with a unique point in \( \mathbb{R}^2 \). This result is significant in topology and differential geometry because it illustrates how topologically non-trivial objects can be analyzed and understood through simpler mathematical structures. The ability to describe the infinite cylinder with a single chart significantly simplifies the analysis of its geometric and topological properties.rcanet62https://www.blogger.com/profile/06232728269658258565noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-54379773744855255502024-02-02T20:15:38.945+01:002024-02-02T20:15:38.945+01:00ChatGPT Answer: “
### Preliminary Concepts
**Man...ChatGPT Answer: “<br /><br />### Preliminary Concepts<br /><br />**Manifold:** An \(n\)-dimensional manifold is a topological space that locally resembles Euclidean space \( \mathbb{R}^n \), but globally can have a more complex topology.<br /><br />**Chart and Atlas:** A chart on a manifold is a homeomorphism from an open subset of the manifold to an open subset of \( \mathbb{R}^n \). An atlas is a collection of charts that cover the entire manifold. If a manifold can be covered by a single chart, this chart provides global coordinates for the manifold.<br /><br />### The Infinite Cylinder \( \mathbb{R} \times S^1 \)<br /><br />The infinite cylinder can be conceptualized as the Cartesian product of the real line \( \mathbb{R} \) and the circle \( S^1 \). The circle can be represented as \( S^1 = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \} \) or, alternatively, through the identification \( S^1 \cong \mathbb{R} / \mathbb{Z} \), where each point on \( S^1 \) corresponds to an equivalence class of real numbers under the equivalence relation \( x \sim y \) if \( x - y \) is an integer.<br /><br />### Explicit Map Construction<br /><br />To cover \( \mathbb{R} \times S^1 \) with a single chart, we need to provide a homeomorphism from this manifold to an open space in \( \mathbb{R}^2 \). A natural way to do this is using angular and radial coordinates, but since the "radius" in our cylinder is constant (for \( S^1 \)), we only need to consider the "height" along \( \mathbb{R} \) and the angle in \( S^1 \).<br /><br />We define the function \( \phi: \mathbb{R} \times S^1 \rightarrow \mathbb{R}^2 \) by:<br />\[ \phi(t, \theta) = (t, \theta) \]<br />where \( t \) is the coordinate along \( \mathbb{R} \), representing the height on the cylinder, and \( \theta \) is the angular coordinate on \( S^1 \), identifying \( S^1 \) with the interval \( [0, 2\pi) \) under the identification of \( 0 \) and \( 2\pi \) as the same point.<br /><br />This function is clearly a homeomorphism, as it is bijective, continuous, and its inverse is also continuous. The continuity and bijectivity come from the direct structure of the Cartesian product and the definition of \( S^1 \) and \( \mathbb{R} \). The inverse \( \phi^{-1}: \mathbb{R}^2 \rightarrow \mathbb{R} \times S^1 \) simply takes a point in \( \mathbb{R}^2 \) and assigns it to its corresponding point on the infinite cylinder.<br /><br />### Conclusion<br /><br />We have shown that the infinite cylinder \( \mathbb{R} \times S^1 \) can be covered by a single chart by explicitly constructing a map \( \phi \) that associates each point on the cylinder with a unique point in \( \mathbb{R}^2 \). This result is significant in topology and differential geometry because it illustrates how topologically non-trivial objects can be analyzed and understood through simpler mathematical structures. The ability to describe the infinite cylinder with a single chart significantly simplifies the analysis of its geometric and topological properties.”rcanet62https://www.blogger.com/profile/06232728269658258565noreply@blogger.com