Canonical connection
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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 $q = gp\in M$ |
$$ X^*_q = \left.\frac{d}{dt}\right|_{t=0} x_tq = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tq) | $$ 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, $$ | = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tgp) = d\pi_g dR_g X, $$ |
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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