Canonical connection
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Let $M$ be a ''homogeneous space'', that is a smooth manifold on which a Lie group $G$ acts transitively by diffeomorphisms. Then $M \cong G/H$ where $H$ is the isotropy group of some base point $p\in M$, and the action map $\pi : G \to M$, $g\mapsto gp$ becomes the canonical projection $\pi : G \to G/H$ which is a [[Principal bundle of smooth manifolds|principal bundle]] with structure group $H$. Let $\textup{Ad} : G \to \Aut(\mathfrak{g})$ denote the adjoint representation of $G$. Its restriction $\textup{Ad}(H)$ clearly keeps $\mathfrak{h}$ invariant. We will assume that the homogeneous space $M = G/H$ is ''reductive'': there is a vector space complement $\mathfrak{m}$ of $\mathfrak{h}$ in $\mathfrak{g}$ which is also invariant under $\textup{Ad}(H)$. Reductiveness is often fulfilled; in particular it holds if $H$ is compact. Since $d\pi_e : \mathfrak{g} \to T_pM$ has kernel $\mathfrak{h}$, it is an isomorphism on the complement $\mathfrak{m}$, and it carries the representation $Ad(H)|_\mathfrak{m}$ into the isotropy representation of $H$ on $T_pM$. Via $d\pi_e$ we identify $\mathfrak{m}$ with $T_pM$. | Let $M$ be a ''homogeneous space'', that is a smooth manifold on which a Lie group $G$ acts transitively by diffeomorphisms. Then $M \cong G/H$ where $H$ is the isotropy group of some base point $p\in M$, and the action map $\pi : G \to M$, $g\mapsto gp$ becomes the canonical projection $\pi : G \to G/H$ which is a [[Principal bundle of smooth manifolds|principal bundle]] with structure group $H$. Let $\textup{Ad} : G \to \Aut(\mathfrak{g})$ denote the adjoint representation of $G$. Its restriction $\textup{Ad}(H)$ clearly keeps $\mathfrak{h}$ invariant. We will assume that the homogeneous space $M = G/H$ is ''reductive'': there is a vector space complement $\mathfrak{m}$ of $\mathfrak{h}$ in $\mathfrak{g}$ which is also invariant under $\textup{Ad}(H)$. Reductiveness is often fulfilled; in particular it holds if $H$ is compact. Since $d\pi_e : \mathfrak{g} \to T_pM$ has kernel $\mathfrak{h}$, it is an isomorphism on the complement $\mathfrak{m}$, and it carries the representation $Ad(H)|_\mathfrak{m}$ into the isotropy representation of $H$ on $T_pM$. Via $d\pi_e$ we identify $\mathfrak{m}$ with $T_pM$. | ||
− | Using left translations $L_g$, $g\in G$, $\mathfrak{m}$ defines a distribution $\mathcal{H}$ on $G$ (``horizontal | + | Using left translations $L_g$, $g\in G$, $\mathfrak{m}$ defines a distribution $\mathcal{H}$ on $G$ (the ``horizontal distribution´´ $\mathcal{H}_g = dL_g\mathfrak{m}$) which is complementary to the vertical distribution $\mathcal{V}_g = dL_g\mathfrak{h}$ and which is invariant under the right translations of $H$ if $\mathfrak{m}$ is a reductive complement. Thus $\mathfrak{m}$ defines a [[Connection on a principal bundle|''connection'']] on the $H$-principal bundle $G \to G/H$, called the ''canonical connection'' of the reductive homogeneous space $(G/H,\mathfrak{m})$. |
The canonical connection determines a covariant derivative $\nabla$ on the tangent bundle $TM$ since this is associated to $\pi:G\to M$, | The canonical connection determines a covariant derivative $\nabla$ on the tangent bundle $TM$ since this is associated to $\pi:G\to M$, | ||
− | $$ (G\times\mathfrak{m})/H \buildrel \cong \over \longrightarrow TM,\ \ \ [g,X] \mapsto d\pi_g dL_g X | + | $$ (G\times\mathfrak{m})/H \buildrel \cong \over \longrightarrow TM,\ \ \ [g,X] \mapsto d\pi_g dL_g X, $$ |
where $h \in H$ acts on $G\times\mathfrak{m}$ by $(g,X) \mapsto (gh^{-1},\textup{Ad}(h)X)$. The covariant derivative $\nabla$ can be defined by its parallel vector fields. A curve $g(t)$ in $G$ is the parallel displacement for $\nabla$ along the path $p(t) = g(t)p$ in $M$ if and only if it is ''horizontal'', $g'(t) \in \mathcal{H}_{g(t)} = dL_{g(t)}\mathfrak{m}$. Thus for every $X \in \mathfrak{m} = T_pM$, the vector field $X(t) = dg(t)_p X$ is parallel along the curve $p(t)$. | where $h \in H$ acts on $G\times\mathfrak{m}$ by $(g,X) \mapsto (gh^{-1},\textup{Ad}(h)X)$. The covariant derivative $\nabla$ can be defined by its parallel vector fields. A curve $g(t)$ in $G$ is the parallel displacement for $\nabla$ along the path $p(t) = g(t)p$ in $M$ if and only if it is ''horizontal'', $g'(t) \in \mathcal{H}_{g(t)} = dL_{g(t)}\mathfrak{m}$. Thus for every $X \in \mathfrak{m} = T_pM$, the vector field $X(t) = dg(t)_p X$ is parallel along the curve $p(t)$. | ||
− | Since $G$ is a transformation group on $M$, its Lie algebra $\mathfrak{g}$ also `` | + | Since $G$ is a transformation group on $M$, its Lie algebra $\mathfrak{g}$ also ``acts´´ on $M$ by the ''action vector fields'': To each $X \in \mathfrak{g}$ we assign a vector field $X^*$ on $M$ by putting for each $q = gp\in M$ |
− | = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tgp) = d\pi_g dR_g X $$ where $t \mapsto x_t$ is a curve in $G$ with $x_0 = e$ and $x'_0 = X$, e.g. | + | $$ X^*_q = \left.\frac{d}{dt}\right|_{t=0} x_tq = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tq) |
+ | = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tgp) = d\pi_g dR_g X, $$ | ||
+ | where $t \mapsto x_t$ is a curve in $G$ with $x_0 = e$ and $x'_0 = X$, e.g. $x_t = \exp(tX)$. Thus $\mathfrak{g}$ is embedded into the Lie algebra of vector fields on $M$. However there is a sign change in the Lie bracket: Note that $X^*$ is $\pi$-related to the '''right''' invariant vector field $X^r$ on $G$ since $X^*_{gp} = d\pi_g dR_g X = d\pi_g X^r_g$; thus $[X^*,Y^*]$ is $\pi$-related to $[X^r,Y^r] = -[X,Y]^r$ and therefore | ||
\begin{equation} \label{action} | \begin{equation} \label{action} | ||
[X^*,Y^*] = -[X,Y]^* . \end{equation} | [X^*,Y^*] = -[X,Y]^* . \end{equation} | ||
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\end{equation} | \end{equation} | ||
From this we may compute the torsion tensor $T$ for any $X,Y \in \mathfrak{m}= T_pM$ by extending $X,Y$ to the action vector fields $X^*,Y^*$ on $M$, using (\ref{action}), (\ref{nabla}): | From this we may compute the torsion tensor $T$ for any $X,Y \in \mathfrak{m}= T_pM$ by extending $X,Y$ to the action vector fields $X^*,Y^*$ on $M$, using (\ref{action}), (\ref{nabla}): | ||
− | \begin{equation} \begin{array} T(X,Y) &=& (\nabla_X Y^* - \nabla_Y X^* - [X^*,Y^*])_p\cr &=& d\pi_e\left(-[X,Y]+[Y,X] + [X,Y]\right) \cr &=& -[X,Y]_\mathfrak{m} \end{array} \end{equation} | + | \begin{equation} \begin{array} T(X,Y) &=& (\nabla_X Y^* - \nabla_Y X^* - [X^*,Y^*])_p\cr &=& d\pi_e\left(-[X,Y]+[Y,X] + [X,Y]\right) \cr &=& -[X,Y]_\mathfrak{m}, \end{array} \end{equation} |
where $Z_\mathfrak{m}$ denotes the $\mathfrak{m}$-component of any $Z\in\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$. | where $Z_\mathfrak{m}$ denotes the $\mathfrak{m}$-component of any $Z\in\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$. | ||
The $\mathfrak{h}$-valued curvature form $R(X,Y) = \Omega(X,Y)$ for $X,Y\in\mathfrak{m}$ is obtained from the [[Connections#Curvature_on_the_frame_bundle|Connections, (7.2)]]: | The $\mathfrak{h}$-valued curvature form $R(X,Y) = \Omega(X,Y)$ for $X,Y\in\mathfrak{m}$ is obtained from the [[Connections#Curvature_on_the_frame_bundle|Connections, (7.2)]]: | ||
− | \begin{equation} R(X,Y) = \Omega[X,Y] = -[X,Y]_\mathfrak{h} \end{equation} | + | \begin{equation} R(X,Y) = \Omega[X,Y] = -[X,Y]_\mathfrak{h}, \end{equation} |
where $Z_\mathfrak{h}$ denotes the $\mathfrak{h}$-component of any $Z\in\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$. Since the connection on $G$ is invariant under left translations, the action of $G$ on $M$ is ''affine'', that is it preserves the covariant derivative $\nabla$, and the same is true for the torsion and the curvature tensors, $T$ and $R$. In particular it is preserved under parallel displacements which are horizontal curves in $G$ (in particular, the holonomy group of $\nabla$ is contained in $H$). Thus these tensors are $\nabla$-parallel. | where $Z_\mathfrak{h}$ denotes the $\mathfrak{h}$-component of any $Z\in\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$. Since the connection on $G$ is invariant under left translations, the action of $G$ on $M$ is ''affine'', that is it preserves the covariant derivative $\nabla$, and the same is true for the torsion and the curvature tensors, $T$ and $R$. In particular it is preserved under parallel displacements which are horizontal curves in $G$ (in particular, the holonomy group of $\nabla$ is contained in $H$). Thus these tensors are $\nabla$-parallel. | ||
− | Vice versa, given any manifold $M$ with a connection $\nabla$ on $TM$ with parallel torsion and curvature tensors, then $M$ is a ''reductive locally homogeneous space'', i.e. each point $p\in M$ has an open neighborhood which can be identified to some set in a reductive homogeneous space $G/H$ where $\nabla$ becomes the canonical connection of $G/H$. In fact, let $\mathfrak{m}= T_pM$ and $H = \Aut(\mathfrak{m},T,R)$ the group of automorphisms of $\mathfrak{m}$ preserving both the `` | + | Vice versa, given any manifold $M$ with a connection $\nabla$ on $TM$ with parallel torsion and curvature tensors, then $M$ is a ''reductive locally homogeneous space'', i.e. each point $p\in M$ has an open neighborhood which can be identified to some set in a reductive homogeneous space $G/H$ where $\nabla$ becomes the canonical connection of $G/H$. In fact, let $\mathfrak{m}= T_pM$ and $H = \Aut(\mathfrak{m},T,R)$ the group of automorphisms of $\mathfrak{m}$ preserving both the ``product´´ $T : \mathfrak{m}\times\mathfrak{m}\to \mathfrak{m}$ and the ``triple product´´ $R : \mathfrak{m}\times\mathfrak{m}\times\mathfrak{m}\to \mathfrak{m}$. Then $G$ is a Lie group with Lie algebra $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ where $\mathfrak{h}\subset\End\mathfrak{m}$ is the Lie algebra of $H$ and where the remaining Lie brackets are given as follows: |
\begin{equation} [A,X] = AX,\ \ \ [X,Y] = -(R(X,Y)+T(X,Y)) \in \mathfrak{h} \oplus \mathfrak{m}= \mathfrak{g} \end{equation} | \begin{equation} [A,X] = AX,\ \ \ [X,Y] = -(R(X,Y)+T(X,Y)) \in \mathfrak{h} \oplus \mathfrak{m}= \mathfrak{g} \end{equation} | ||
for all $A\in\mathfrak{h}$ and $X,Y\in\mathfrak{m}$. | for all $A\in\mathfrak{h}$ and $X,Y\in\mathfrak{m}$. | ||
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<wikitex>; | <wikitex>; | ||
Let $M$ be the set of all real positive definite symmetric $n\times n$-matrices which is an open subset of the space of all symmetric matrices, $S_n = \{X\in\R^{n\times n}: X^t = X\}$. The group $GL_n$ acts transitively on $M$ by $(g,p) \mapsto gpg^t$. The isotropy group of $p = I$ (unit matrix) is the orthogonal group $H = O_n = \{g\in\R^{n\times n}: g^tg = I\}$ whose Lie algebra is the space of antisymmetric matrices, $A_n = | Let $M$ be the set of all real positive definite symmetric $n\times n$-matrices which is an open subset of the space of all symmetric matrices, $S_n = \{X\in\R^{n\times n}: X^t = X\}$. The group $GL_n$ acts transitively on $M$ by $(g,p) \mapsto gpg^t$. The isotropy group of $p = I$ (unit matrix) is the orthogonal group $H = O_n = \{g\in\R^{n\times n}: g^tg = I\}$ whose Lie algebra is the space of antisymmetric matrices, $A_n = | ||
− | \{A\in \R^{n\times n}: A^t = -A\}$. Thus $M \cong GL_n/O_n$, and the reductive splitting is $\mathfrak{g} = \R^{n\times n} = A_n \oplus S_n$ with $A_n = \mathfrak{h}$ and $S_n = \mathfrak{m}$. The vector space $S_n$ is not a Lie algebra since for the commutator of any $X,Y \in S_n$ we have $[X,Y]\in A_n$. But it is a ''Lie triple'': Note that $[A,X] \in S_n$ for all $A\in A_n$, $X \in S_n$ (since $(AX-XA)^t = X^tA^t-A^tX^t = -XA+AX$), in particular $[[X,Y],Z] \in S_n$ for all $X,Y,Z\in S_n$. The canonical connection is torsion free (i.e. | + | \{A\in \R^{n\times n}: A^t = -A\}$. Thus $M \cong GL_n/O_n$, and the reductive splitting is $\mathfrak{g} = \R^{n\times n} = A_n \oplus S_n$ with $A_n = \mathfrak{h}$ and $S_n = \mathfrak{m}$. The vector space $S_n$ is not a Lie algebra since for the commutator of any $X,Y \in S_n$ we have $[X,Y]\in A_n$. But it is a ''Lie triple'': Note that $[A,X] \in S_n$ for all $A\in A_n$, $X \in S_n$ (since $(AX-XA)^t = X^tA^t-A^tX^t = -XA+AX$), in particular $[[X,Y],Z] \in S_n$ for all $X,Y,Z\in S_n$. The canonical connection is torsion free (i.e. $M$ is symmetric) since $[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$, and its curvature tensor is $R(X,Y)Z = - [[X,Y],Z]$. The trace metric on $S_n$ is invariant under $H = O_n$ and extends to a $GL_n$-invariant Riemannian metric on $M$ given as follows: For any $p\in M$ and $X,Y \in T_pM = S_n$ we put $$ \langle X,Y\rangle _p := \textup{trace} p^{-1}Xp^{-1}Y $$ Thus the canonical connection is the Levi-Civita connection on $M$ with respect to this metric. The sectional curvature is nonpositive; it is given by $\langle R(X,Y)Y,X\rangle = -\textup{trace}([[X,Y],Y]X) = \textup{trace}[X,Y]^2 \leq 0$ (recall that $[X,Y] \in A_n$ has imaginary eigenvalues, hence its square has nonpositive trace). |
+ | |||
+ | For further information, see \cite{Kowalski1980}. | ||
</wikitex> | </wikitex> | ||
== References == | == References == |
Latest revision as of 11:40, 21 May 2013
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1 Definition
Let be a homogeneous space, that is a smooth manifold on which a Lie group acts transitively by diffeomorphisms. Then where is the isotropy group of some base point , and the action map , becomes the canonical projection which is a principal bundle with structure group . Let denote the adjoint representation of . Its restriction clearly keeps invariant. We will assume that the homogeneous space is reductive: there is a vector space complement of in which is also invariant under . Reductiveness is often fulfilled; in particular it holds if is compact. Since has kernel , it is an isomorphism on the complement , and it carries the representation into the isotropy representation of on . Via we identify with .
Using left translations , , defines a distribution on (the ``horizontal distribution´´ ) which is complementary to the vertical distribution and which is invariant under the right translations of if is a reductive complement. Thus defines a connection on the -principal bundle , called the canonical connection of the reductive homogeneous space .
The canonical connection determines a covariant derivative on the tangent bundle since this is associated to ,
where acts on by . The covariant derivative can be defined by its parallel vector fields. A curve in is the parallel displacement for along the path in if and only if it is horizontal, . Thus for every , the vector field is parallel along the curve .
Since is a transformation group on , its Lie algebra also ``acts´´ on by the action vector fields: To each we assign a vector field on by putting for each
where is a curve in with and , e.g. . Thus is embedded into the Lie algebra of vector fields on . However there is a sign change in the Lie bracket: Note that is -related to the right invariant vector field on since ; thus is -related to and therefore
From this we may compute the torsion tensor for any by extending to the action vector fields on , using (1), (2):
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where denotes the -component of any .
The -valued curvature form for is obtained from the Connections, (7.2):
where denotes the -component of any . Since the connection on is invariant under left translations, the action of on is affine, that is it preserves the covariant derivative , and the same is true for the torsion and the curvature tensors, and . In particular it is preserved under parallel displacements which are horizontal curves in (in particular, the holonomy group of is contained in ). Thus these tensors are -parallel.
Vice versa, given any manifold with a connection on with parallel torsion and curvature tensors, then is a reductive locally homogeneous space, i.e. each point has an open neighborhood which can be identified to some set in a reductive homogeneous space where becomes the canonical connection of . In fact, let and the group of automorphisms of preserving both the ``product´´ and the ``triple product´´ . Then is a Lie group with Lie algebra where is the Lie algebra of and where the remaining Lie brackets are given as follows:
for all and .
When , these spaces are called locally symmetric. This happens if and only if the torsion tensor vanishes. Moreover, if carries a -invariant (semi-)Riemannian metric, the canonical connection preserves the metric and is torsion free, hence it is the Levi-Civita connection of .
2 Example
For further information, see [Kowalski1980].
3 References
- [Kowalski1980] O. Kowalski, Generalized symmetric spaces, LNM 805, Springer-Verlag, 1980. MR579184 (83d:53036) Zbl 0614.53040