# Homogeneous spaces

### Smooth Manifold

An n-dimensional manifold is a second countable Hausdorff topological space $M$ such that

• $M$ is locally Euclidean: $\forall p \in M,$ there is $U_p \in \tau(p)$ and a diffeomorphism $\phi_p:U_p \rightarrow \widetilde{U}_p \subset \mathbb{R}^n.$ $(U,\phi)$ is a coordinate system.
• $M$ has a differentiable structure: there is a collection $\mathcal{U}:= \{ (U_{\alpha}) \}_{\alpha \in I}$ of coordinate systems such that:
a. $M = \cup_{\alpha \in I} U_{\alpha}$.
b. $\phi_{\alpha}\circ \phi_{\beta}^{-1}$ is $C^{\infty}$.
c. $\mathcal{U}$ is maximal with respect to (b): if $(U, \phi)$ is a c.s. such that $\phi_{\alpha}\circ \phi^{-1}$ and $\phi\circ \phi_{\alpha}^{-1}$ are smooth, then $(U,\phi) \in \mathcal{U}$.

### Lie groups

• Lie Group = Smooth manifold + Topological Group
• $U(m)= \{ m \times m \,\,\, \textit{complex matrices such that} \,\,\,A^{*}=A^{-1} \}$ $SO(n)=\{ A \in \mathbb{R}^{n\times n}: det A =1 \}$
• $SU(m)=\{ A \in U(m): det A =1 \}$

### Homogeneous manifolds

• Let $G$ be a Lie group. If $K$ is an abstract closed subgroup of $G$, with the relative topology $K$ is a Lie subgroup of $G$ ($K$ is a Lie group, $\exists \phi \in Hom(K,G)$), $(\phi, K)$ is a sub-manifold of $G$: $G: d\phi_k:T_kK \rightarrow T_{\phi(k)}G$ is non-singular $\forall k \in K,$ and $\phi$ is $1 - 1$.
• Let $G$ be a Lie group. If $K$ is closed subgroup of $G,$ let $M = G/H = \{ gK : g \in G \}$. Let $p:G \rightarrow G/K$ the natural projection: $p(g)=gK$. Then, $M$ has a unique manifold structure such
a. $p$ is $C^{\infty}$.
b. There exists local sections of $G/K$ in $G$: if $gK\in M$, then there exists $W \in \tau(gK)$ and $s\in C^{\infty}(W,G)$, such that $p\circ s= id_M$.
$M$ above satisfying (a) and (b) is called a homogeneous manifold.

Action of a Lie group on a smooth manifold
An action of $G$ on $M$ is a $C^{\infty}$ – map $\cdot : G \times M \rightarrow M$ such that

• $\forall p \in M, \,\,\, \forall h,g \in G, \,\,\, h\cdot (g\cdot p) = hg \cdot p$,
• $\forall p \in M, \,\,\, e_G \cdot p =p.$
• Effective, if $\forall p,q \in M$, there exists $g=g(p,q)\in G$ such that $g \cdot p=q$.
• Transitive, if

Transitive actions of Lie groups on manifolds
Let $\cdot : G \times M \rightarrow M$ be a transitive action and let $p \in M$.

• $K = G_p:= \{ g \in G: g \cdot p =p \}$ is a closed subgroup of $G$, called the isotropy subgroup at $p$.
• The mapping $\beta: G / G_p \rightarrow M, \,\,\, gG_p \mapsto g\cdot p,$ is a diffeomorphism.
• $M\cong G / G_p, \,\,\, \forall p \in M$.

Examples:

• $G = SO(n). \,\,\, p \in \mathbb{S}^{n-1}.$
$SO(n) \times \mathbb{S}^{n-1} \rightarrow \mathbb{S}^{n-1}$
$G_p = SO(n)_p= \{ M \in SO(n) : M \cdot p=p \}\cong SO(n-1) .$
$SO(n) / SO(n-1) \cong \mathbb{S}^{n-1} .$
• $G = U(n). \,\,\, p \in \mathbb{S}^{2n-1}.$
$U(n) \times \mathbb{S}^{2n-1} \rightarrow \mathbb{S}^{2n-1}$
$G_p = U(n)_p= \{ M \in U(n) : M \cdot p=p \}\cong U(n-1) .$
$U(n) / U(n-1) \cong \mathbb{S}^{2n-1} .$
• $G = SU(n). \,\,\, p \in \mathbb{S}^{2n-1}.$
$SU(n) \times \mathbb{S}^{2n-1} \rightarrow \mathbb{S}^{2n-1}$
$G_p = SU(n)_p= \{ M \in SU(n) : M \cdot p=p \}\cong SU(n-1) .$
$SU(n) / SU(n-1) \cong \mathbb{S}^{2n-1} \cong \mathbb{C}\mathbb{P}^{n-1}.$
• $SU(2) \cong SU(2) / SU(1) \cong \mathbb{S}^{3} .$
• $SU(2)/ \mathbb{T}^1 \cong \mathbb{S}^2 .$
$\mathbb{T}^1= \{ diag[ e^{2\pi i \theta}, e^{-2\pi \theta}]: \theta \in [0,1] \} .$

Hopf fibration:

• $SU(2)/ \mathbb{T}^1 \cong \mathbb{S}^2 .$
• $\mathbb{SU}(2) \cong \mathbb{S}^3, \,\,\, \mathbb{T}^1 \cong \mathbb{S}^1 .$
• $\mathbb{S}^3/ \mathbb{S}^1 \cong \mathbb{S}^2 .$

Fibration $I = [0,1] .$

A continuous surjection $p: E \rightarrow F$ is a fibration, if $\forall X, (X,p)$ has the Homotopy lifting property:
$\forall f: X \times I \rightarrow B,$
$\forall \widetilde{f}_0 : X \times I \rightarrow E$ lifting $f_0 = f|_{X\times\{0\}}$,
– There exists $\widetilde{f}:X \times I \rightarrow E$ $f$ to $E$: $f=p\circ \widetilde{f}$.

Hopf Fibration.

There is a continuous surjection $p \mathbb{S}^3 \rightarrow \mathbb{S}^2$ such that $\forall X, \,\,\, (X,p)$ has the Homopoty lifting property:

## Vector Bundles.$\mathbb{K} = \mathbb{R},\mathbb{C}.$

A $C^{\infty} \,\,\,\,\, \mathbb{K}$– vector bundle is a triple $(p,E,M),$ where $p : E \rightarrow M$ is a $C^{\infty}$ surjection such that:
(i) $\forall x \in M, \,\,\, E_x = p^{-1}(x)$ is a $\mathbb{K}$-vector space.
(ii) $\forall x \in M, \exists U \in \tau(x),$ and a homeomorphism $\phi : E_U:= p^{-1}(U) \rightarrow U \times \mathbb{K}^n$ is a linear mapping; (ii) says that $E$ is trivial in $U$.
(iii) There exists an open covering $\mathfrak{U} = \{ U_i\}_{i\in I}$ of $M: \forall x \in M,$ there exists $U_i$, such that $x \in U_i \subset M,$ with $E$ being trivial in $U_i$.

Examples:
– Trivial vector bundles: $E = B\times \mathbb{K}^n, \,\,\, p:E \rightarrow B, \,\,\, p(b,v)=b.$
– Tangent bundle: $TM \rightarrow M.$

Examples:
Line bundle on the real projective plane $\mathbb{R}\mathbb{P}^n.$
$u,v \in \mathbb{S}^n, \,\,\, u \sim v \Leftrightarrow u=-v. \,\,\, \mathbb{R}\mathbb{P}^n = \mathbb{S}^n/ \sim.$
$\gamma_n = \{ (x,l):x \in \mathbb{R}\mathbb{P}^n, l \in \mathbb{R}u\}.$
$p: \gamma_n \rightarrow \mathbb{R}\mathbb{P}^n$ is called the line bundle of $\mathbb{R}\mathbb{P}^n.$
$\gamma_1 \rightarrow \mathbb{R}\mathbb{P}^1 = \mathbb{S}^1$ is the infinite Möbius band.