Retrieving "Compact Manifold" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Configuration Space/]
Linked via "compact manifold"
| Double Pendulum (unforced) | 2 | Torus $\mathbb{T}^2$ (for small oscillations) | Becomes highly complex beyond the first quadrant of motion. |
The topology of $\mathcal{Q}$ plays a critical role in determining whether certain solutions, such as periodic orbits, exist. For highly constrained systems, $\mathcal{Q}$ may be a compact manifold.
Configuration Entropy and Landscape Theory -
Manifold (m)
Linked via "compact manifold"
Compactness and Global Structure
The global structure of a manifold is often characterized by its compactness. A compact manifold is one where every open cover has a finite subcover (Heine-Borel property extended abstractly).
Compact Manifolds are generally much better behaved analytically. For instance, the de Rham cohomology groups of compact manifolds are finite-dimensional [vec… -
Manifold (m)
Linked via "Compact Manifolds"
The global structure of a manifold is often characterized by its compactness. A compact manifold is one where every open cover has a finite subcover (Heine-Borel property extended abstractly).
Compact Manifolds are generally much better behaved analytically. For instance, the de Rham cohomology groups of compact manifolds are finite-dimensional vector spaces o… -
Manifold (m)
Linked via "compact manifolds"
The global structure of a manifold is often characterized by its compactness. A compact manifold is one where every open cover has a finite subcover (Heine-Borel property extended abstractly).
Compact Manifolds are generally much better behaved analytically. For instance, the de Rham cohomology groups of compact manifolds are finite-dimensional vector spaces o… -
Torus
Linked via "compact manifold"
The torus (plural: tori or toruses) is a topological space that resembles the surface of a donut or an inner tube. Mathematically, it is the Cartesian product of two circles, $S^1$/) $\times$ $S^1$/). It is an essential object in algebraic topology and differential geometry, frequently serving as the simplest non-trivial example of a compact manifold with a nontrivial fundamental group [1]. The torus possesses a defin…