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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
Dec 11, 2016 · A manifold is a curved space that is locally flat. Think of the surface of the Earth, which is a two-dimensional manifold (can be described using two coordinates - latitude and longitude). Small patches of the Earth's surface can be described using Euclidean geometry; bigger areas can't as this geometry breaks down.
This page titled 5.11: Manifolds (Part 1) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Benjamin Crowell via source content that was edited to the style and standards of the LibreTexts platform. General relativity doesn’t assume a predefined background metric, and this creates a chicken-and-egg problem.
Oct 11, 2015 · A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.If you want to lear...
- 4 min
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- GeometryForPhysicists
The set of all smooth scalar fields on a manifold will be denoted by. F(M) = {f : M → R|f smooth}. (10) The coordinate functions xi : U → R (the i-th coordinate in a chart on M) are like scalar fields, but they are only defined locally (inside the chart U). For many purposes, they can be treated as scalar fields.
We discuss the idea of manifolds informally, and then give a formal definition, discussing the underlying concepts of topological spaces, continuous maps and...
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- Richard Southwell
This page titled 5.12: Manifolds (Part 2) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Benjamin Crowell via source content that was edited to the style and standards of the LibreTexts platform. An alternative way of characterizing an n-manifold is as an object that can locally be described by n real ...
