Difference between revisions of "Manifold"
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Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]]. | Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]]. | ||
| − | Example: Consider the set of points <math>[x_0:x_1:x_2]</math>, where <math>x_i</math> are not all zero, and where we declare <math>[x_0:x_1:x_2]</math> to be equivalent to <math>[y_0:y_1:y_2]</math> if <math>x_i = \lambda y_i</math>. This space describes the set <math>\mathbf{R}P^2</math> of all one-dimensional subspaces of <math>R^3</math>. | + | Example: Consider the set of points <math>[x_0:x_1:x_2]</math>, where <math>x_i</math> are not all zero, and where we declare <math>[x_0:x_1:x_2]</math> to be equivalent to <math>[y_0:y_1:y_2]</math> if <math>x_i = \lambda y_i</math>. This space describes the set <math>\mathbf{R}P^2</math> of all one-dimensional subspaces of <math>R^3</math>. Let <math>U_i</math> be the open set with <math>x_i \neq 0</math>. Define <math>\phi_0: U_0\rightarrow R^2</math> to be the map given by <math>[x_0:x_1:x_2]\mapsto (\frac{x_1}{x_0},\frac{x_2}{x_0})</math>. Define <math>\phi_1</math> and <math>\phi_2</math> similarly. Then <math>\phi_i</math> clearly define homeomorphisms from <math>U_i</math> to <math>R^2</math>. It is simple to verify that the change of coordinate functions are also diffeomorphisms. Hence <math>\mathbf{R}P^2</math> is a differentiable manifold. In fact, <math>\mathbf{R}P^2</math> with a disk removed actually defines the mobius strip. |
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
[[category: Topology]] | [[category: Topology]] | ||
Revision as of 22:26, June 30, 2008
An n-dimensional manifold (or n-manifold) M is a topological space such that every point in M has a neighbourhood U that is homeomorphic to Rn. The homeomorphisms should be thought of as providing local coordinates in the neighborhood U. Whenever two such coordinate neighborhoods U and V intersect, we also require that the change of coordinate maps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_U\circ\phi_V^{-1}:\phi_V(U\cap V)\rightarrow \phi_U(U\cap V)} be homeomorphisms.
An n-dimensional differentiable manifold is an n-manifold with the property that the change of coordinate maps are differentiable functions.
An n-dimensional complex manifold N is a topological space such that every point in N has a neighbourhood that is homeomorphic to Cn, and such that the change of coordinate functions are holomorphic.
Manifolds are Hausdorff and 2nd-countable.
Example: Consider the set of points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_0:x_1:x_2]} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} are not all zero, and where we declare Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_0:x_1:x_2]} to be equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [y_0:y_1:y_2]} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i = \lambda y_i} . This space describes the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}P^2} of all one-dimensional subspaces of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^3} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} be the open set with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i \neq 0} . Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_0: U_0\rightarrow R^2} to be the map given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_0:x_1:x_2]\mapsto (\frac{x_1}{x_0},\frac{x_2}{x_0})} . Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_2} similarly. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i} clearly define homeomorphisms from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^2} . It is simple to verify that the change of coordinate functions are also diffeomorphisms. Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}P^2} is a differentiable manifold. In fact, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}P^2} with a disk removed actually defines the mobius strip.