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Manifold

2026-01-22T13:14:53Z

PrincipalBundle: /* Definitions */

== Definitions ==
A topological manifold of dimension $n$ is a topological space that is Hausdorff, second countable and locally Euclidean (i.e. every point has a neighbourhood homeomorphic to $\mathbb{R}^n$.

A smooth atlas on a topological manifold is an open cover $M=\bigcup_\alpha U_\alpha$ with associated maps (called charts) $\varphi_\alpha: U_\alpha \to \varphi_\alpha(U_\alpha) \subset \mathbb{R}^n$, where the image $\varphi_\alpha(U_\alpha)$ is open in $\mathbb{R}^n$ and $\varphi_\alpha$ is a homeomorphism onto its image. These charts must be compatible: whenever $U_\alpha \cap U_\beta \neq \emptyset$, the transition function

<math> \varphi_\beta \circ \varphi_\alpha^{-1}: \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)</math>

is a diffeomorphism.

A smooth manifold is a topological manifold with a choice of smooth atlas.

Two atlases are compatible if their union is an atlas.

Manifold

2026-01-22T13:10:24Z

PrincipalBundle: /* Definition */

== Definitions ==
A topological manifold of dimension $n$ is a topological space that is Hausdorff, second countable and locally Euclidean (i.e. every point has a neighbourhood homeomorphic to $\mathbb{R}^n$.

A smooth atlas on a topological manifold is an open cover $M=\bigcup_\alpha U_\alpha$ with associated maps (called charts) $\varphi_\alpha: U_\alpha \to \varphi_\alpha(U_\alpha) \subset \mathbb{R}^n$, where the image $\varphi_\alpha(U_\alpha)$ is open in $\mathbb{R}^n$ and $\varphi_\alpha$ is a homeomorphism onto its image. These charts must be compatible: whenever $U_\alpha \cap U_\beta \neq \emptyset$, the transition function

$\varphi_\beta \circ \varphi_\alpha^{-1}: \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)$

is a diffeomorphism.

A smooth manifold is a topological manifold with a choice of smooth atlas.

Two atlases are compatible if their union is an atlas.

Lie Group

2026-01-22T12:58:38Z

PrincipalBundle: /* Definition */

== Definition ==

A group G, is a Lie group if G is a [[Manifold | smooth manifold]] such that group multiplication $m:G\times G\to G$ and inversion $i:G\to G$ are smooth maps.

On every Lie group there are maps $L_a, R_a, C_a$ defined by left multiplying by $a$, right multiplying by $a$, and conjugation by $a$ ($x\mapsto axa^{-1}$) respectively. The maps $L_a$ and $R_a$ are diffeomorphisms.

== Morphisms ==

A morphism of Lie groups $\phi:G\to H$ is a smooth group homomorphism. A Lie group morphism is an isomorphism if it is a diffeomorphism that is also a group homomorphism.

Lie groups and Lie group morphisms form a category.

== Examples ==

Let $V$ be a vector space. Then $(V, +)$ is a Lie group.

$(\Z, +)$ is a $0$-dimensional Lie group. Similarly, any countable group with the discrete topology is a $0$-dimensional Lie group.

$\R / \Z$ is a $1$-dimensional Lie group.

$S^3 = \{(x_1, y_1, x_2, y_2) \in \R^4: \Vert-\Vert = 1\}$ with group structure $(z_1, z_2) \cdot (z_3, z_4) := (z_1 z_3 - z_2 \overline{z_4}, z_1 z_4 - z_2 \overline{z_3})$ where we treat $\R^4$ as $\C^2$.

=== Matrix Lie Groups ===

For $k=\R$ or $k=\C$

$GL_n(k)$ invertible matrices with entries in $k$.

$SL_n(k)$ matrices with determinant 1.

$O_n(k)$ matricies such that $AA^T=A^TA=1$.

Lie Group

2026-01-22T12:55:51Z

PrincipalBundle: /* Examples */

== Definition ==

A group G, is a Lie group if G is a smooth manifold such that group multiplication $m:G\times G\to G$ and inversion $i:G\to G$ are smooth maps.

On every Lie group there are maps $L_a, R_a, C_a$ defined by left multiplying by $a$, right multiplying by $a$, and conjugation by $a$ ($x\mapsto axa^{-1}$) respectively. The maps $L_a$ and $R_a$ are diffeomorphisms.

== Morphisms ==

A morphism of Lie groups $\phi:G\to H$ is a smooth group homomorphism. A Lie group morphism is an isomorphism if it is a diffeomorphism that is also a group homomorphism.

Lie groups and Lie group morphisms form a category.

== Examples ==

Let $V$ be a vector space. Then $(V, +)$ is a Lie group.

$(\Z, +)$ is a $0$-dimensional Lie group. Similarly, any countable group with the discrete topology is a $0$-dimensional Lie group.

$\R / \Z$ is a $1$-dimensional Lie group.

$S^3 = \{(x_1, y_1, x_2, y_2) \in \R^4: \Vert-\Vert = 1\}$ with group structure $(z_1, z_2) \cdot (z_3, z_4) := (z_1 z_3 - z_2 \overline{z_4}, z_1 z_4 - z_2 \overline{z_3})$ where we treat $\R^4$ as $\C^2$.

=== Matrix Lie Groups ===

For $k=\R$ or $k=\C$

$GL_n(k)$ invertible matrices with entries in $k$.

$SL_n(k)$ matrices with determinant 1.

$O_n(k)$ matricies such that $AA^T=A^TA=1$.