Manifold (geometry): Difference between revisions
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Differentiable manifolds have a [[tangent space]] <math>T_p M</math>, the space of all [[tangent vector]]s, associated with each point <math>p</math> on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within <math>\scriptstyle \mathbb{R}^{n+1}</math>, it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible. | Differentiable manifolds have a [[tangent space]] <math>T_p M</math>, the space of all [[tangent vector]]s, associated with each point <math>p</math> on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within <math>\scriptstyle \mathbb{R}^{n+1}</math>, it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible. | ||
===Riemannian | ===Riemannian manifold=== | ||
To define distances and angles on a differentiable manifold, it is necessary to define a '''[[metric]]'''. A differentiable manifold equipped with a metric is called a '''Riemannian manifold'''. A Riemannian metric is a generalisation of the usual idea of the scalar or [[dot product]] to a manifold. In other words, a Riemannian metric <math> g = \{g_p\}_{p \in M} </math> is a set of symmetric [[inner product]]s | To define distances and angles on a differentiable manifold, it is necessary to define a '''[[metric]]'''. A differentiable manifold equipped with a metric is called a '''Riemannian manifold'''. A Riemannian metric is a generalisation of the usual idea of the scalar or [[dot product]] to a manifold. In other words, a Riemannian metric <math> g = \{g_p\}_{p \in M} </math> is a set of symmetric [[inner product]]s | ||
Revision as of 06:32, 22 July 2007
A manifold is an abstract mathematical space that looks locally like Euclidean space, but globally may have a very different structure. An example of this is a sphere: if one is very close to the surface of the sphere, it looks like a flat plane, but globally the sphere and plane are very different. Other examples of manifolds include lines and circles, and more abstract spaces such as the orthogonal group O(n).
The concept of a manifold is very important within mathematics and physics, and is fundamental to certain fields such as differential geometry, Riemannian geometry and general relativity.
The most basic manifold is a topological manifold, but additional structures can be defined on the manifold to create objects such as differentiable manifolds and Riemannian manifolds.
Mathematical definition
Topological manifold
In topology, a manifold of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , or an n-manifold, is defined as a Hausdorff space where an open neighbourhood of each point is homeomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^n } (i.e. there exists a smooth bijective map from the manifold with a smooth inverse to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^n } ).
Differentiable manifold
To define differentiable manifolds, the concept of an atlas, chart and a coordinate change need to be introduced. An atlas of the Earth uses these concepts: the atlas is a collection of different overlapping patches of small parts of a spherical object onto a plane. The way in which these different patches overlap is defined by the coordinate change.
Let M be a set. An atlas of M is a collection of pairs Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(U_{\alpha}, \psi_{\alpha}\right)} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha} varying over an index set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } such that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{\alpha} \in M, \quad M = \bigcup_{\alpha \in A} U_{\alpha} }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{\alpha} } maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{\alpha}} bijectively to an open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle V_{\alpha} \in \, \mathbb{R}^n} , and for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha,\,\beta \,\in A } the image Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \psi(U_{\alpha} \cap U_{\beta}) \, \in \, \mathbb{R}^n} is an open set. The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{\alpha}: U_{\alpha} \rightarrow V_{\alpha} } is called a chart.
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha, \, \beta \, \in A} , the coordinate change is a differentiable map between two open sets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^n } whereby Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \qquad \psi_{\beta} \circ \psi_{\alpha}^{-1}: \psi_{\alpha} \left(U_{\alpha} \cap U_{\beta} \right) \rightarrow \psi_{\beta} (U_{\alpha} \cap U_{\beta}).}
The set M is a differentiable manifold if and only if it comes equipped with a countable atlas and satisfies the Hausdorff property. The important difference between a differentiable manifold and a topological manifold is that the charts are diffeomorphisms (a differentiable function with a differentiable inverse) rather than homeomorphisms.
Differentiable manifolds have a tangent space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p M} , the space of all tangent vectors, associated with each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^{n+1}} , it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible.
Riemannian manifold
To define distances and angles on a differentiable manifold, it is necessary to define a metric. A differentiable manifold equipped with a metric is called a Riemannian manifold. A Riemannian metric is a generalisation of the usual idea of the scalar or dot product to a manifold. In other words, a Riemannian metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g = \{g_p\}_{p \in M} } is a set of symmetric inner products
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_p : T_pM \times T_pM \rightarrow \mathbb{R} }
which depend smoothly on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} .
Examples of manifolds
This list is not exhaustive.
- Circles, lines, planes etc.
- Paraboloid of revolution
- Real projective space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle R \mathbb{P}^2} -- the set of all lines through the origin in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^n }
- The Grassmannian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle G_k(\mathbb{R}^n)} -- The set of all k-dimensional subspaces of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}^n} .
- Lie groups e.g. O(n), U(n), GL(n),</math> and the Lorentz group,