Canonical connection

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1 Definition

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a homogeneous space, that is a smooth manifold on which a Lie group $G$$G$ acts transitively by diffeomorphisms. Then $M \cong G/H$$M \cong G/H$ where $H$$H$ is the isotropy group of some base point $p\in M$$p\in M$, and the action map $\pi : G \to M$$\pi : G \to M$, $g\mapsto gp$$g\mapsto gp$ becomes the canonical projection $\pi : G \to G/H$$\pi : G \to G/H$ which is a principal bundle with structure group $H$$H$. Let $\textup{Ad} : G \to \Aut(\mathfrak{g})$$\textup{Ad} : G \to \Aut(\mathfrak{g})$ denote the adjoint representation of $G$$G$. Its restriction $\textup{Ad}(H)$$\textup{Ad}(H)$ clearly keeps $\mathfrak{h}$$\mathfrak{h}$ invariant. We will assume that the homogeneous space $M = G/H$$M = G/H$ is reductive: there is a vector space complement $\mathfrak{m}$$\mathfrak{m}$ of $\mathfrak{h}$$\mathfrak{h}$ in $\mathfrak{g}$$\mathfrak{g}$ which is also invariant under $\textup{Ad}(H)$$\textup{Ad}(H)$. Reductiveness is often fulfilled; in particular it holds if $H$$H$ is compact. Since $d\pi_e : \mathfrak{g} \to T_pM$$d\pi_e : \mathfrak{g} \to T_pM$ has kernel $\mathfrak{h}$$\mathfrak{h}$, it is an isomorphism on the complement $\mathfrak{m}$$\mathfrak{m}$, and it carries the representation $Ad(H)|_\mathfrak{m}$$Ad(H)|_\mathfrak{m}$ into the isotropy representation of $H$$H$ on $T_pM$$T_pM$. Via $d\pi_e$$d\pi_e$ we identify $\mathfrak{m}$$\mathfrak{m}$ with $T_pM$$T_pM$.

Using left translations $L_g$$L_g$, $g\in G$$g\in G$, $\mathfrak{m}$$\mathfrak{m}$ defines a distribution $\mathcal{H}$$\mathcal{H}$ on $G$$G$ (the ``horizontal distribution´´ $\mathcal{H}_g = dL_g\mathfrak{m}$$\mathcal{H}_g = dL_g\mathfrak{m}$) which is complementary to the vertical distribution $\mathcal{V}_g = dL_g\mathfrak{h}$$\mathcal{V}_g = dL_g\mathfrak{h}$ and which is invariant under the right translations of $H$$H$ if $\mathfrak{m}$$\mathfrak{m}$ is a reductive complement. Thus $\mathfrak{m}$$\mathfrak{m}$ defines a connection on the $H$$H$-principal bundle $G \to G/H$$G \to G/H$, called the canonical connection of the reductive homogeneous space $(G/H,\mathfrak{m})$$(G/H,\mathfrak{m})$.

The canonical connection determines a covariant derivative $\nabla$$\nabla$ on the tangent bundle $TM$$TM$ since this is associated to $\pi:G\to M$$\pi:G\to M$,

$\displaystyle (G\times\mathfrak{m})/H \buildrel \cong \over \longrightarrow TM,\ \ \ [g,X] \mapsto d\pi_g dL_g X,$ $$\displaystyle (G\times\mathfrak{m})/H \buildrel \cong \over \longrightarrow TM,\ \ \ [g,X] \mapsto d\pi_g dL_g X,$$

where $h \in H$$h \in H$ acts on $G\times\mathfrak{m}$$G\times\mathfrak{m}$ by $(g,X) \mapsto (gh^{-1},\textup{Ad}(h)X)$$(g,X) \mapsto (gh^{-1},\textup{Ad}(h)X)$. The covariant derivative $\nabla$$\nabla$ can be defined by its parallel vector fields. A curve $g(t)$$g(t)$ in $G$$G$ is the parallel displacement for $\nabla$$\nabla$ along the path $p(t) = g(t)p$$p(t) = g(t)p$ in $M$$M$ if and only if it is horizontal, $g'(t) \in \mathcal{H}_{g(t)} = dL_{g(t)}\mathfrak{m}$$g'(t) \in \mathcal{H}_{g(t)} = dL_{g(t)}\mathfrak{m}$. Thus for every $X \in \mathfrak{m} = T_pM$$X \in \mathfrak{m} = T_pM$, the vector field $X(t) = dg(t)_p X$$X(t) = dg(t)_p X$ is parallel along the curve $p(t)$$p(t)$.

Since $G$$G$ is a transformation group on $M$$M$, its Lie algebra $\mathfrak{g}$$\mathfrak{g}$ also ``acts´´ on $M$$M$ by the action vector fields: To each $X \in \mathfrak{g}$$X \in \mathfrak{g}$ we assign a vector field $X^*$$X^*$ on $M$$M$ by putting for each $q = gp\in M$$q = gp\in M$

$\displaystyle X^*_q = \left.\frac{d}{dt}\right|_{t=0} x_tq = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tq) = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tgp) = d\pi_g dR_g X,$ $$\displaystyle X^*_q = \left.\frac{d}{dt}\right|_{t=0} x_tq = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tq) = \left.\frac{d}{dt}\right|_{t=0} \pi(x_tgp) = d\pi_g dR_g X,$$

where $t \mapsto x_t$$t \mapsto x_t$ is a curve in $G$$G$ with $x_0 = e$$x_0 = e$ and $x'_0 = X$$x'_0 = X$, e.g. $x_t = \exp(tX)$$x_t = \exp(tX)$. Thus $\mathfrak{g}$$\mathfrak{g}$ is embedded into the Lie algebra of vector fields on $M$$M$. However there is a sign change in the Lie bracket: Note that $X^*$$X^*$ is $\pi$$\pi$-related to the right invariant vector field $X^r$$X^r$ on $G$$G$ since $X^*_{gp} = d\pi_g dR_g X = d\pi_g X^r_g$$X^*_{gp} = d\pi_g dR_g X = d\pi_g X^r_g$; thus $[X^*,Y^*]$$[X^*,Y^*]$ is $\pi$$\pi$-related to $[X^r,Y^r] = -[X,Y]^r$$[X^r,Y^r] = -[X,Y]^r$ and therefore

(1)$[X^*,Y^*] = -[X,Y]^* .$$[X^*,Y^*] = -[X,Y]^* .$

Now we can compute $\nabla_{X} Y^*$$\nabla_{X} Y^*$ for any $Y\in \mathfrak{g}$$Y\in \mathfrak{g}$ and $X \in \mathfrak{m}= T_pM$$X \in \mathfrak{m}= T_pM$. Putting $g_t = \exp(tX)$$g_t = \exp(tX)$ and $Y(t) = \textup{Ad}(g_t^{-1})Y$$Y(t) = \textup{Ad}(g_t^{-1})Y$ we have

$\displaystyle Y^*_{g_tp} = d\pi_{g_t} dR_{g_t}Y = d\pi_{g_t} dL_{g_t}Y(t) = (dg_t)_p d\pi_e Y(t).$ $$\displaystyle Y^*_{g_tp} = d\pi_{g_t} dR_{g_t}Y = d\pi_{g_t} dL_{g_t}Y(t) = (dg_t)_p d\pi_e Y(t).$$

Since $g_t$$g_t$ is the parallel transport along the curve $t\mapsto g_tp$$t\mapsto g_tp$ in $M$$M$, we obtain $\nabla_{g_t'} Y^* = dg_td\pi_e Y'(t).$$\nabla_{g_t'} Y^* = dg_td\pi_e Y'(t).$ Thus for $t=0$$t=0$ we have

(2)$\nabla_X Y^* = d\pi_e Y'(0) = -d\pi_e[X,Y].$$\nabla_X Y^* = d\pi_e Y'(0) = -d\pi_e[X,Y].$

From this we may compute the torsion tensor $T$$T$ for any $X,Y \in \mathfrak{m}= T_pM$$X,Y \in \mathfrak{m}= T_pM$ by extending $X,Y$$X,Y$ to the action vector fields $X^*,Y^*$$X^*,Y^*$ on $M$$M$, using (1), (2):

(3)

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$\begin{array} T(X,Y) &=& (\nabla_X Y^* - \nabla_Y X^* - [X^*,Y^*])_p\cr &=& d\pi_e\left(-[X,Y]+[Y,X] + [X,Y]\right) \cr &=& -[X,Y]_\mathfrak{m}, \end{array}$

where $Z_\mathfrak{m}$$Z_\mathfrak{m}$ denotes the $\mathfrak{m}$$\mathfrak{m}$-component of any $Z\in\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$$Z\in\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$.

The $\mathfrak{h}$$\mathfrak{h}$-valued curvature form $R(X,Y) = \Omega(X,Y)$$R(X,Y) = \Omega(X,Y)$ for $X,Y\in\mathfrak{m}$$X,Y\in\mathfrak{m}$ is obtained from the Connections, (7.2):

(4)$R(X,Y) = \Omega[X,Y] = -[X,Y]_\mathfrak{h},$$R(X,Y) = \Omega[X,Y] = -[X,Y]_\mathfrak{h},$

where $Z_\mathfrak{h}$$Z_\mathfrak{h}$ denotes the $\mathfrak{h}$$\mathfrak{h}$-component of any $Z\in\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$$Z\in\mathfrak{g} = \mathfrak{h}\oplus\mathfrak{m}$. Since the connection on $G$$G$ is invariant under left translations, the action of $G$$G$ on $M$$M$ is affine, that is it preserves the covariant derivative $\nabla$$\nabla$, and the same is true for the torsion and the curvature tensors, $T$$T$ and $R$$R$. In particular it is preserved under parallel displacements which are horizontal curves in $G$$G$ (in particular, the holonomy group of $\nabla$$\nabla$ is contained in $H$$H$). Thus these tensors are $\nabla$$\nabla$-parallel.

Vice versa, given any manifold $M$$M$ with a connection $\nabla$$\nabla$ on $TM$$TM$ with parallel torsion and curvature tensors, then $M$$M$ is a reductive locally homogeneous space, i.e. each point $p\in M$$p\in M$ has an open neighborhood which can be identified to some set in a reductive homogeneous space $G/H$$G/H$ where $\nabla$$\nabla$ becomes the canonical connection of $G/H$$G/H$. In fact, let $\mathfrak{m}= T_pM$$\mathfrak{m}= T_pM$ and $H = \Aut(\mathfrak{m},T,R)$$H = \Aut(\mathfrak{m},T,R)$ the group of automorphisms of $\mathfrak{m}$$\mathfrak{m}$ preserving both the ``product´´ $T : \mathfrak{m}\times\mathfrak{m}\to \mathfrak{m}$$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}$$R : \mathfrak{m}\times\mathfrak{m}\times\mathfrak{m}\to \mathfrak{m}$. Then $G$$G$ is a Lie group with Lie algebra $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$$\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ where $\mathfrak{h}\subset\End\mathfrak{m}$$\mathfrak{h}\subset\End\mathfrak{m}$ is the Lie algebra of $H$$H$ and where the remaining Lie brackets are given as follows:

(5)$[A,X] = AX,\ \ \ [X,Y] = -(R(X,Y)+T(X,Y)) \in \mathfrak{h} \oplus \mathfrak{m}= \mathfrak{g}$$[A,X] = AX,\ \ \ [X,Y] = -(R(X,Y)+T(X,Y)) \in \mathfrak{h} \oplus \mathfrak{m}= \mathfrak{g}$

for all $A\in\mathfrak{h}$$A\in\mathfrak{h}$ and $X,Y\in\mathfrak{m}$$X,Y\in\mathfrak{m}$.

When $[\mathfrak{m},\mathfrak{m}] \subset \mathfrak{h}$$[\mathfrak{m},\mathfrak{m}] \subset \mathfrak{h}$, these spaces are called locally symmetric. This happens if and only if the torsion tensor $T = [\mathfrak{m},\mathfrak{m}]_\mathfrak{m}$$T = [\mathfrak{m},\mathfrak{m}]_\mathfrak{m}$ vanishes. Moreover, if $\tilde M$$\tilde M$ carries a $G$$G$-invariant (semi-)Riemannian metric, the canonical connection $\nabla$$\nabla$ preserves the metric and is torsion free, hence it is the Levi-Civita connection of $\tilde M$$\tilde M$.

2 Example

Let $M$$M$ be the set of all real positive definite symmetric $n\times n$$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\}$$S_n = \{X\in\R^{n\times n}: X^t = X\}$. The group $GL_n$$GL_n$ acts transitively on $M$$M$ by $(g,p) \mapsto gpg^t$$(g,p) \mapsto gpg^t$. The isotropy group of $p = I$$p = I$ (unit matrix) is the orthogonal group $H = O_n = \{g\in\R^{n\times n}: g^tg = I\}$$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\}$$A_n = \{A\in \R^{n\times n}: A^t = -A\}$. Thus $M \cong GL_n/O_n$$M \cong GL_n/O_n$, and the reductive splitting is $\mathfrak{g} = \R^{n\times n} = A_n \oplus S_n$$\mathfrak{g} = \R^{n\times n} = A_n \oplus S_n$ with $A_n = \mathfrak{h}$$A_n = \mathfrak{h}$ and $S_n = \mathfrak{m}$$S_n = \mathfrak{m}$. The vector space $S_n$$S_n$ is not a Lie algebra since for the commutator of any $X,Y \in S_n$$X,Y \in S_n$ we have $[X,Y]\in A_n$$[X,Y]\in A_n$. But it is a Lie triple: Note that $[A,X] \in S_n$$[A,X] \in S_n$ for all $A\in A_n$$A\in A_n$, $X \in S_n$$X \in S_n$ (since $(AX-XA)^t = X^tA^t-A^tX^t = -XA+AX$$(AX-XA)^t = X^tA^t-A^tX^t = -XA+AX$), in particular $[[X,Y],Z] \in S_n$$[[X,Y],Z] \in S_n$ for all $X,Y,Z\in S_n$$X,Y,Z\in S_n$. The canonical connection is torsion free (i.e. $M$$M$ is symmetric) since $[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$$[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$, and its curvature tensor is $R(X,Y)Z = - [[X,Y],Z]$$R(X,Y)Z = - [[X,Y],Z]$. The trace metric on $S_n$$S_n$ is invariant under $H = O_n$$H = O_n$ and extends to a $GL_n$$GL_n$-invariant Riemannian metric on $M$$M$ given as follows: For any $p\in M$$p\in M$ and $X,Y \in T_pM = S_n$$X,Y \in T_pM = S_n$ we put

$\displaystyle \langle X,Y\rangle _p := \textup{trace} p^{-1}Xp^{-1}Y$ $$\displaystyle \langle X,Y\rangle _p := \textup{trace} p^{-1}Xp^{-1}Y$$

Thus the canonical connection is the Levi-Civita connection on $M$$M$ with respect to this metric. The sectional curvature is nonpositive; it is given by $\langle R(X,Y)Y,X\rangle = -\textup{trace}([[X,Y],Y]X) = \textup{trace}[X,Y]^2 \leq 0$$\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$$[X,Y] \in A_n$ has imaginary eigenvalues, hence its square has nonpositive trace).

For further information, see [Kowalski1980].

3 References

• [Kowalski1980] O. Kowalski, Generalized symmetric spaces, LNM 805, Springer-Verlag, 1980. MR579184 (83d:53036) Zbl 0614.53040