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 $\pi:G\to M$, |
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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