Manifold (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-07

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Manifold


00:00:37 1 Motivating examples
00:01:14 1.1 Circle
00:03:26 1.2 Sphere
00:03:44 1.3 Enriched circle
00:04:22 1.4 Other curves
00:04:59 2 Mathematical definition
00:05:37 3 Charts, atlases, and transition maps
00:06:14 3.1 Charts
00:06:52 3.2 Atlases
00:08:07 3.3 Transition maps
00:08:44 3.4 Additional structure
00:09:22 4 Manifold with boundary
00:09:59 4.1 Boundary and interior
00:10:37 5 Construction
00:11:14 5.1 Charts
00:11:52 5.1.1 Sphere with charts
00:12:29 5.2 Patchwork
00:12:48 5.2.1 Intrinsic and extrinsic view
00:13:25 5.2.2 in/i-Sphere as a patchwork
00:14:03 5.3 Identifying points of a manifold
00:14:40 5.4 Gluing along boundaries
00:15:18 5.5 Cartesian products
00:15:55 6 History
00:16:33 6.1 Early development
00:17:10 6.2 Synthesis
00:17:48 6.3 Poincaré's definition
00:18:25 6.4 Topology of manifolds: highlights
00:19:03 7 Additional structure
00:19:40 7.1 Topological manifolds
00:20:18 7.2 Differentiable manifolds
00:20:55 7.3 Riemannian manifolds
00:21:33 7.4 Finsler manifolds
00:22:10 7.5 Lie groups
00:22:29 7.6 Other types of manifolds
00:23:07 8 Classification and invariants
00:23:44 9 Surfaces
00:24:40 9.1 Orientability
00:25:18 9.1.1 Möbius strip
00:25:37 9.1.2 Klein bottle
00:26:14 9.1.3 Real projective plane
00:26:51 9.2 Genus and the Euler characteristic
00:27:29 10 Maps of manifolds
00:28:06 10.1 Scalar-valued functions
00:28:44 11 Generalizations of manifolds
00:29:21 12 See also
00:29:59 12.1 By dimension
00:30:18 13 Notes
00:30:55 14 References
00:31:33 15 External links
00:32:10 Real projective plane
00:32:48 Genus and the Euler characteristic
00:33:25 Maps of manifolds
00:34:03 Scalar-valued functions
00:34:40 Generalizations of manifolds
00:35:55 See also
00:36:14 By dimension



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SUMMARY
=======
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.  More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.
One-dimensional manifolds include lines and circles, but not figure eights (because no neighborhood of their crossing point is homeomorphic to Euclidean 1-space).  Two-dimensional manifolds are also called surfaces.  Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may be not homeomorphic to Euclidean space.  For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts).  When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space.  Manifolds naturally arise as solution sets of systems of equations and as graphs of functions.
Manifolds can be equipped with additional structure.  One important class of manifolds is the class of d ...