03.03.07 · differential-geometry / lie-groups

Invariant affine connections on a reductive homogeneous space and the canonical connection

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Anchor (Master): Kobayashi-Nomizu Vol. II Ch. X §2-§4 (the canonical connection, parallel curvature and torsion, completeness); Nomizu 1954 (Amer. J. Math. 76, 33-65); Helgason Ch. IV §§1-3

Intuition Beginner

A homogeneous space is a space that looks the same when viewed from any of its points. The sphere is one: a symmetry of the sphere can slide any point to any other point, so no point is special. The round globe, a flat plane, the saddle-shaped hyperbolic plane — all of these have a symmetry group large enough to move any point onto any other.

On such a space we want a rule for transporting a tangent arrow along a path while keeping its direction "constant." That rule is a connection. The natural demand is that the rule respect the symmetries: if you transport an arrow and then slide the whole picture by a symmetry, you should get the same answer as sliding first and transporting after. A connection that passes this test is called invariant.

The surprise is that all such symmetry-respecting rules can be listed by a single piece of bookkeeping at one point. You do not have to check the whole space. You only describe how the rule behaves in the tangent directions at the basepoint, and the symmetries copy that behaviour everywhere else.

Among all invariant rules there is a most economical one, called the canonical connection. It transports arrows along the natural "straight" paths traced by the symmetry group itself, adding nothing extra.

Visual Beginner

Alt text: On the left, a sphere shows a tangent arrow at the north pole carried down a meridian to the equator by the rotation symmetry, ending as a definite arrow. On the right, the same sphere is first rotated by a symmetry and the arrow is then carried by the rule; the final arrow matches the one on the left. The matching pictures show that an invariant connection gives the same answer whether you apply the symmetry before or after transporting, so the whole rule is fixed by its behaviour in the tangent directions at a single basepoint. An inset marks those tangent directions as the data the symmetry group spreads over the space.

Worked example Beginner

Take the round sphere, thought of as all the positions a rotation can move the north pole to. The rotations that fix the north pole form a smaller group: spinning the sphere about its vertical axis. The tangent plane at the north pole is where the action happens, and a vertical spin rotates that tangent plane.

Now pick the canonical rule. To carry an arrow from the north pole down a meridian, ride the rotation that slides the pole down that meridian and let it carry the arrow along. This is the most natural transport possible: the symmetry does all the work, and the rule adds nothing.

This rule produces the meridians themselves as the "straight" paths. A bug walking a meridian at steady speed feels it is going straight, because the carrying rotation never tells it to turn. So the canonical connection makes the orbits of the symmetry group into its straight lines, and on the sphere those orbits are the great circles through the pole.

Check your understanding Beginner

Formal definition Intermediate+

Let $G$ be a Lie group with closed subgroup $H$ , and let $M = G / H$ with basepoint $o = eH$ . The natural left action of $G$ on $M$ makes $G$ a transitive group of diffeomorphisms; write $τ_{g} : M \to M$ for the action of $g$ . Let $g$ and $h$ be the Lie algebras of $G$ and $H$ . The pair $(G / H)$ is reductive if there is a vector-space complement $m$ to $h$ in $g$ , $$ \mathfrak g = \mathfrak h \oplus \mathfrak m, \qquad \mathrm{Ad}(H),\mathfrak m \subseteq \mathfrak m, $$ that is invariant under the adjoint action of $H$ 03.02.37. Differentiating $τ$ at $o$ identifies $m$ with the tangent space $T_{o} M$ , and under this identification the isotropy representation of $H$ on $T_{o} M$ becomes $Ad (H) ∣_{m}$ . For $X \in g$ write $X^{*}$ for the induced fundamental vector field on $M$ , $X_{p}^{*} = \frac{d}{d t}_{0} exp (tX) \cdot p$ .

A $G$ -invariant affine connection on $M$ is an affine connection $\nabla$ (equivalently, a connection in the linear frame bundle 03.05.15) such that each $τ_{g}$ is an affine transformation: $τ_{g}$ commutes with $\nabla$ . The structural object that classifies these is purely algebraic.

Definition (Nomizu map). A Nomizu map is a bilinear map $$ \alpha : \mathfrak m \times \mathfrak m \to \mathfrak m $$ that is $Ad (H)$ -equivariant, meaning $α (Ad (h) X, Ad (h) Y) = Ad (h) α (X, Y)$ for all $h \in H$ and $X, Y \in m$ . When $H$ is connected this is the infinitesimal condition $$ \alpha\big([Z,X]{\mathfrak m},, Y\big) + \alpha\big(X,, [Z,Y]{\mathfrak m}\big) = [Z,, \alpha(X,Y)]{\mathfrak m} \quad\text{for } Z \in \mathfrak h . $$ Here, for $W \in g$ , $W{\mathfrak h} $an d$ W_{\mathfrak m} $d e n o t e t h eco m p o n e n t s in t h er e d u c t i v es pl i tt in g . B ec a u se$ \mathrm{Ad}(h) $p r eser v es$ \mathfrak m $, t h e b r a c k e t$ [Z, X] $f or$ Z \in \mathfrak h $,$ X \in \mathfrak m $a l r e a d y l i es in$ \mathfrak m $, so n o p r o j ec t i o ni s n ee d e d in t ha tt er m; t h e p r o j ec t i o n$ [\cdot,\cdot]_{\mathfrak m} $ma tt er so n l y f or b r a c k e t so f tw o$ \mathfrak m$-elements.

Definition (canonical connection). The canonical connection of the reductive space $(G / H, m)$ is the invariant connection corresponding to the zero Nomizu map $α \equiv 0$ . The natural torsion-free connection is the one corresponding to $$ \alpha(X,Y) = \tfrac12,[X,Y]{\mathfrak m}, \qquad X, Y \in \mathfrak m, $$ which is $Ad (H)$ -equivariant because $[\cdot,\cdot]{\mathfrak m} $an d$ \mathrm{Ad}(H) $co mm u t eo n$ \mathfrak m$.

The covariant derivative of an invariant connection along the orbit directions reads, at the basepoint, $$ (\nabla_{X^} Y^)o = \alpha(X, Y) - \tfrac12,[X,Y]{\mathfrak m} \quad\text{for } X, Y \in \mathfrak m, $$ a normalisation in which the canonical connection ( $α = 0$ ) and the natural torsion-free connection ( $α = \frac{1}{2} [\cdot, \cdot]_{m}$ ) are read off directly. (Conventions differ across texts by the placement of the $\frac{1}{2}$ ; we fix Nomizu's, so that $α = 0$ is the canonical connection.)

Counterexamples to common slips

Reductive is a choice, not a property. A complement $m$ with $Ad (H) m \subseteq m$ need not be unique; different reductive complements give different canonical connections. When $H$ is compact one may average to produce one, but the canonical connection depends on the chosen $m$ .
Invariant connection versus invariant metric. An invariant metric always yields an invariant connection (its Levi-Civita connection), but most invariant connections do not come from any invariant metric; the Nomizu maps form a whole vector space while metric-compatible ones are a thin slice of it.
The canonical connection is rarely torsion-free. Its torsion at $o$ is $T (X, Y) = - [X, Y]_{m}$ , which vanishes only when $[m, m] \subseteq h$ — the symmetric-space condition. On a generic reductive space the canonical connection has torsion.

Key theorem with proof Intermediate+

Theorem (Nomizu). Let $G / H$ be a reductive homogeneous space with fixed reductive decomposition $g = h \oplus m$ . Then there is a bijection $$ {,G\text{-invariant affine connections on } G/H,} ;\longleftrightarrow; {,\mathrm{Ad}(H)\text{-equivariant bilinear maps } \alpha : \mathfrak m \times \mathfrak m \to \mathfrak m,}. $$ Under this bijection the canonical connection ( $α = 0$ ) has the following properties at the basepoint $o$ . Its torsion and curvature are $$ T(X,Y) = -[X,Y]{\mathfrak m}, \qquad R(X,Y) = -\mathrm{ad}\big([X,Y]{\mathfrak h}\big)\big|_{\mathfrak m}, \qquad X, Y \in \mathfrak m, $$ both invariant under $G$ ; its geodesics through $o$ are the orbit curves $t \mapsto exp (tX) \cdot o$ for $X \in m$ ; it is complete; and its torsion and curvature are parallel, $$ \nabla T = 0, \qquad \nabla R = 0 . $$

Proof. An invariant connection in the frame bundle is determined by an $Ad (H)$ -equivariant linear map $Λ_{m} : g \to gl (m)$ describing the connection form along the orbit fields; the constraint that it be a connection forces $Λ_{m} (Z) = ad (Z) ∣_{m}$ for $Z \in h$ , so the only free data is the restriction to $m$ , which is exactly a bilinear $α (X, Y) := Λ_{m} (X) Y$ with $X, Y \in m$ . Equivariance of $Λ_{m}$ becomes equivariance of $α$ . This is the Nomizu correspondence ^{[Kobayashi-Nomizu Ch. X §2 Thm 2.1]}.

Set $α = 0$ . Torsion is $T (X^{*}, Y^{*}) = \nabla_{X^{*}} Y^{*} - \nabla_{Y^{*}} X^{*} - [X^{*}, Y^{*}]$ . The fundamental-field bracket satisfies $[X^{*}, Y^{*}] = - [X, Y]^{*}$ , and at $o$ the field $W^{*}$ has value $W_{m}$ under $T_{o} M ≅ m$ . With $α = 0$ the formula $(\nabla_{X^{*}} Y^{*})_{o} = - \frac{1}{2} [X, Y]_{m}$ gives $$ T(X,Y) = -\tfrac12[X,Y]{\mathfrak m} + \tfrac12[Y,X]{\mathfrak m} + [X,Y]{\mathfrak m} = -[X,Y]{\mathfrak m}. $$ For curvature one computes $R (X^{*}, Y^{*}) Z^{*}$ at $o$ ; the $m$ -brackets cancel through the Jacobi identity and only the $h$ -component survives, acting by its adjoint, giving $R (X, Y) = - ad ([X, Y]_{h}) ∣_{m}$ ^{[Nomizu 1954]}.

The orbit curve $γ (t) = exp (tX) \cdot o$ has velocity the invariant field generated by $X$ ; since $\nabla_{X^{*}} X^{*} = α (X, X) - \frac{1}{2} [X, X]_{m} = 0$ for $α = 0$ , the curve is auto-parallel, hence a geodesic. These curves are defined for all $t \in R$ , so the connection is complete. Finally $T$ and $R$ are built from $G$ -invariant data ( $Ad (H)$ -equivariant brackets) and are therefore $G$ -invariant tensor fields; a $G$ -invariant tensor on a homogeneous space is parallel for the canonical connection precisely because parallel transport along orbit geodesics is realised by the group action, giving $\nabla T = 0$ and $\nabla R = 0$ ^{[Kobayashi-Nomizu Ch. X §2 Thm 2.10]}. $■$

Bridge. This theorem builds toward the entire structure theory of symmetric and naturally reductive spaces, and the canonical connection appears again in 07.04.07, where the symmetric-space condition $[m, m] \subseteq h$ collapses the torsion to zero and the curvature to $R (X, Y) = - ad ([X, Y])$ . The foundational reason the whole classification reduces to algebra is the Nomizu bijection: an analytic object (a connection on a possibly large manifold) is exactly an equivariant bilinear map on a single vector space, and this is exactly the move that turns invariant geometry into representation theory. Putting these together, the canonical connection generalises the flat connection on a Lie group (the case $H = {e}$ , $m = g$ , where $T = - [\cdot, \cdot]$ recovers the Cartan $(-)$ -connection), and the central insight — that $\nabla R = 0$ and $\nabla T = 0$ characterise the homogeneous geometries — is dual to the Ambrose-Singer holonomy reconstruction of 03.05.13, reading the same curvature data forward instead of backward.

Exercises Intermediate+

Exercise (medium, short-answer). Suppose the isotropy representation $\mathrm{Ad}(H)|_{\mathfrak m}$ is irreducible. Using Schur's lemma, describe the space of $\mathrm{Ad}(H)$-invariant symmetric bilinear forms on $\mathfrak m$, and explain what this says about invariant metrics.

Hint

A real irreducible representation has endomorphism algebra a division ring; for the typical (real type) case it is $R$ .

Answer

One-dimensional (real type). If $Ad (H) ∣_{m}$ is irreducible of real type, Schur forces the invariant inner products to be scalar multiples of a single one, so there is a unique invariant metric up to scale. Such an isotropy-irreducible space is automatically Einstein. The canonical connection, however, exists regardless of any metric; the bijection of the theorem is about connections, and the metric question is the narrower symmetric-bilinear-form slice studied in 03.02.37.

Exercise (hard, short-answer). Prove that for the canonical connection, a $G$-invariant tensor field on $G/H$ is parallel. (This gives $\nabla T = 0$, $\nabla R = 0$ as special cases.)

Hint

Parallel transport along the geodesic $exp (tX) \cdot o$ is realised by the differential of $τ_{e x p (tX)}$ . A $G$ -invariant tensor is fixed by every $τ_{g}$ .

Answer

Rubric. For the canonical connection, parallel transport along the orbit geodesic $t \mapsto exp (tX) \cdot o$ coincides with the differential of the diffeomorphism $τ_{e x p (tX)}$ (both are the group action carrying $o$ along the curve, and $α = 0$ means no extra rotation is added). A $G$ -invariant tensor $S$ satisfies $(τ_{g})_{*} S = S$ , so its parallel transport returns $S$ unchanged; hence $\nabla_{\overset{γ}{˙}} S = 0$ along every orbit geodesic. Since orbit geodesics through $o$ span every tangent direction, $\nabla S = 0$ at $o$ , and by homogeneity everywhere. Full credit identifies parallel transport with the group action and uses invariance of $S$ .

Exercise (hard, short-answer). Show that the canonical connection and the natural torsion-free connection have the same geodesics through $o$, but differ in torsion. Why is this consistent?

Hint

Two connections share geodesics iff their difference tensor $D (X, Y) = α_{1} (X, Y) - α_{2} (X, Y)$ is antisymmetric; geodesics see only the symmetric part.

Answer

They share geodesics. The difference of the two Nomizu maps is $α_{tf} - α_{can} = \frac{1}{2} [\cdot, \cdot]_{m}$ , which is antisymmetric in $X, Y$ . Geodesics depend only on the symmetric part $\frac{1}{2} (\nabla_{X} Y + \nabla_{Y} X)$ , and an antisymmetric difference tensor contributes nothing there, so both have the orbit curves $exp (tX) \cdot o$ as geodesics. Torsion, by contrast, is built from the antisymmetric part, so the two connections differ exactly in torsion: the canonical one carries $T = - [\cdot, \cdot]_{m}$ , the natural one carries $T = 0$ . Two connections with the same geodesics and different torsion is the standard phenomenon of projectively related connections.

Advanced results Master

Naturally reductive spaces and the metric canonical connection. A reductive homogeneous space with an invariant metric $⟨ \cdot, \cdot ⟩$ on $m$ is naturally reductive when $⟨[X, Y]_{m}, Z ⟩ + ⟨ Y, [X, Z]_{m} ⟩ = 0$ for all $X, Y, Z \in m$ . In that case the canonical connection is metric (parallelises the invariant metric) and its geodesics through $o$ are again the orbit curves $exp (tX) \cdot o$ , so they coincide with the Riemannian geodesics of the Levi-Civita connection through $o$ . The Levi-Civita and canonical connections then differ by the torsion tensor $- \frac{1}{2} [\cdot, \cdot]_{m}$ , an invariant totally skew tensor, exhibiting the canonical connection as a metric connection with skew torsion. This is the bridge between the algebraic canonical connection and Riemannian geometry, and it is exactly the structure used to compute homogeneous Einstein metrics in 03.02.37.

The symmetric-space specialisation. When $[m, m] \subseteq h$ the space is (locally) symmetric. The torsion $T = - [\cdot, \cdot]_{m}$ vanishes, so the canonical connection is torsion-free; being also $G$ -invariant and metric for any invariant metric, it is the Levi-Civita connection of that metric. The curvature collapses to $R (X, Y) = - ad ([X, Y]) ∣_{m}$ with $[X, Y] \in h$ , the formula derived independently in 07.04.07 from the geodesic symmetry $s_{o}$ . Here $\nabla R = 0$ is automatic from the general parallel-curvature statement, recovering the defining property of a locally symmetric space. The geodesic symmetry $s_{o} (exp X \cdot o) = exp (- X) \cdot o$ is the affine reflection that the canonical connection makes an affine transformation.

Holonomy and the de Rham reconstruction. For the canonical connection the holonomy algebra at $o$ is the smallest subalgebra of $gl (m)$ containing all curvature operators $R (X, Y) = - ad ([X, Y]_{h}) ∣_{m}$ , which is contained in $ad (h) ∣_{m}$ . Because $\nabla R = 0$ , the Ambrose-Singer construction terminates at this single layer: no higher covariant derivatives of curvature contribute. This makes the canonical connection the cleanest setting in which to see holonomy as a Lie-algebraic shadow of the isotropy representation.

The affine holonomy and Nomizu's completeness theorem. Nomizu proved more: a complete simply-connected affine manifold with $\nabla T = 0$ and $\nabla R = 0$ is reductive homogeneous with its canonical connection, the converse direction. Parallel torsion and curvature are not merely consequences of homogeneity — they characterise it. This is the affine analogue of the Cartan-Ambrose-Hicks recognition of symmetric spaces by $\nabla R = 0$ , and it places the canonical connection at the centre of the structure theory.

Synthesis. The canonical connection putting these together is the central insight that organises invariant differential geometry: a single equivariant bilinear map on $m$ encodes a connection, and the most economical choice $α = 0$ produces torsion and curvature that are pure brackets, manifestly parallel. This is exactly the structure that the symmetric-space theory of 07.04.07 specialises, the naturally reductive metric theory of 03.02.37 enriches, and the holonomy theory builds toward. It generalises the flat $(-)$ -connection of a Lie group and is dual to the Levi-Civita connection in the precise sense that the two differ by an invariant skew torsion tensor; the foundational reason the entire homogeneous catalogue is computable is that $\nabla T = 0$ and $\nabla R = 0$ turn global geometry into a finite computation in $g$ . The bridge from the analytic to the algebraic is the Nomizu bijection, and every later homogeneous-geometry result reads off this one correspondence.

Full proof set Master

Proposition (the curvature of the canonical connection). At the basepoint $o$ of a reductive homogeneous space with the canonical connection, $R (X, Y) Z = - [[X, Y]_{h}, Z]$ for $X, Y, Z \in m$ , where $[\cdot, \cdot]_{h}$ is the $h$ -component of the bracket.

Proof. The curvature of an invariant connection with Nomizu map $α$ is, at $o$ , $$ R(X,Y)Z = \alpha(X)\alpha(Y)Z - \alpha(Y)\alpha(X)Z - \alpha\big([X,Y]{\mathfrak m}\big)Z - \big[[X,Y]{\mathfrak h},, Z\big], $$ writing $α (X)$ for the operator $Z \mapsto α (X, Z)$ . The last term is the contribution of the $h$ -bracket through the forced value $Λ_{m} (Z) = ad (Z) ∣_{m}$ on $h$ . Setting $α = 0$ kills the first three terms and leaves $R (X, Y) Z = - [[X, Y]_{h}, Z]$ , i.e. $R (X, Y) = - ad ([X, Y]_{h}) ∣_{m}$ . The bracket $[[X, Y]_{h}, Z]$ lies in $m$ because $[h, m] \subseteq m$ , so the operator preserves $m$ as required. $■$

Proposition (geodesics are orbit curves). For the canonical connection the geodesics through $o$ are exactly $γ_{X} (t) = exp (tX) \cdot o$ , $X \in m$ , and the connection is complete.

Proof. The velocity of $γ_{X}$ is the value at $γ_{X} (t)$ of the field generated by $X$ , transported by the action. The geodesic equation at $o$ requires $\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0$ ; along an orbit this reduces to $α (X, X) - \frac{1}{2} [X, X]_{m} = α (X, X)$ , which is $0$ for the canonical connection since $α = 0$ . Invariance under $τ_{e x p (s X)}$ propagates the auto-parallel condition along the whole curve, so $γ_{X}$ is a geodesic. Each $γ_{X}$ is defined for all $t \in R$ because $exp (tX)$ is, so every maximal geodesic through $o$ — and by homogeneity through any point — is complete. $■$

Proposition (parallel curvature and torsion). For the canonical connection, $\nabla T = 0$ and $\nabla R = 0$ .

Proof. Both $T (X, Y) = - [X, Y]_{m}$ and $R (X, Y) = - ad ([X, Y]_{h}) ∣_{m}$ are $Ad (H)$ -equivariant tensors on $m$ and hence define $G$ -invariant tensor fields on $G / H$ . Along the geodesic $exp (tX) \cdot o$ parallel transport for the canonical connection equals $d τ_{e x p (tX)}$ (Exercise: hard, short-answer). A $G$ -invariant tensor is fixed by $d τ_{g}$ , so its covariant derivative along every orbit geodesic vanishes; as these geodesics realise every tangent direction at $o$ , the full covariant derivative vanishes at $o$ , and by homogeneity everywhere. Hence $\nabla T = 0$ and $\nabla R = 0$ . $■$

The converse — that a complete simply-connected affine manifold with $\nabla T = 0$ , $\nabla R = 0$ is reductive homogeneous with this connection — is Nomizu's recognition theorem, stated above; the proof reconstructs $G$ from the affine transvection (transport) group ^{[Nomizu 1954]}.

Connections Master

Reductive decomposition and invariant metrics 03.02.37. That unit supplies the splitting $g = h \oplus m$ with $Ad (H) m \subseteq m$ and the isotropy representation that this unit's Nomizu maps are equivariant for. Where 03.02.37 classifies invariant metrics (the symmetric-bilinear-form slice) and their Levi-Civita curvature, this unit classifies invariant connections (the full equivariant-bilinear-map space). The naturally reductive case is exactly where the two meet: the canonical connection becomes metric with skew torsion.
Riemannian symmetric space 07.04.07. The symmetric-space condition $[m, m] \subseteq h$ is the special case $α = 0$ with vanishing torsion; there the canonical connection is the Levi-Civita connection and $R (X, Y) = - ad ([X, Y])$ , $\nabla R = 0$ are recovered. That unit derives these from the geodesic symmetry; this unit derives them as a corollary of the general reductive canonical connection, so the two are the same geometry approached from opposite ends.
Linear connection, frame bundle, soldering form 03.05.15. The invariant affine connection is a connection in the linear frame bundle invariant under the $G$ -action; the torsion here is the torsion of 03.05.15 computed through the soldering form, and the Nomizu map is the frame-bundle connection form restricted to the reductive complement.
Associated bundle and induced connection 03.05.13. The canonical connection is induced on every associated bundle of the principal $H$ -bundle $G \to G / H$ by the reductive splitting viewed as a principal connection on $G \to G / H$ ; the Ambrose-Singer holonomy reconstruction of curvature terminates at one layer here because $\nabla R = 0$ .
Geodesics and parallel transport 13.02.02. The geodesics of the canonical connection are the orbit curves $exp (tX) \cdot o$ , and parallel transport along them is realised by the group action — the cleanest instance of the parallel-transport machinery built there.

Historical & philosophical context Master

The theory originates with Katsumi Nomizu's 1954 paper "Invariant affine connections on homogeneous spaces" (American Journal of Mathematics 76, 33-65), which established the bijection between invariant connections and equivariant bilinear maps and singled out the canonical connection with its parallel torsion and curvature ^{[Nomizu 1954]}. The paper grew out of Élie Cartan's programme on symmetric spaces and the Cartan-Schouten connections on Lie groups (the $(+)$ , $(-)$ , and $(0)$ connections of 1926), of which the canonical connection is the homogeneous-space generalisation. Nomizu's recognition theorem — that completeness together with $\nabla T = 0$ and $\nabla R = 0$ characterises reductive homogeneous spaces — reframed homogeneity itself as a curvature condition, paralleling Cartan's earlier characterisation of symmetric spaces by $\nabla R = 0$ .

The synthesis that made the theory canonical is Chapter X of Kobayashi and Nomizu's Foundations of Differential Geometry, Volume II (1969), which recast the construction in the frame-bundle language of principal connections ^{[Kobayashi-Nomizu Ch. X §2 Thm 2.1]}. Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces (1978) gave the now-standard textbook treatment, embedding the canonical connection in the theory of affine and Riemannian symmetric spaces ^{[Helgason Ch. IV §1-§3]}. Philosophically, the result is an early and clean instance of a recurring theme: a geometric structure invariant under a transitive symmetry group is determined by finite linear-algebraic data at one point, so global differential geometry becomes representation theory of the isotropy group.

Bibliography Master

@article{Nomizu1954Invariant,
  author  = {Nomizu, Katsumi},
  title   = {Invariant affine connections on homogeneous spaces},
  journal = {American Journal of Mathematics},
  volume  = {76},
  pages   = {33--65},
  year    = {1954}
}

@book{KobayashiNomizu1969Vol2,
  author    = {Kobayashi, Shoshichi and Nomizu, Katsumi},
  title     = {Foundations of Differential Geometry, Volume II},
  series    = {Interscience Tracts in Pure and Applied Mathematics},
  number    = {15},
  publisher = {Interscience Publishers (John Wiley \& Sons)},
  year      = {1969},
  note      = {Chapter X: Homogeneous spaces and the canonical connection}
}

@book{Helgason1978,
  author    = {Helgason, Sigurdur},
  title     = {Differential Geometry, Lie Groups, and Symmetric Spaces},
  publisher = {Academic Press},
  year      = {1978},
  note      = {Chapter IV: invariant connections and the canonical connection}
}

@article{Cartan1926,
  author  = {Cartan, {\'E}lie},
  title   = {Sur une classe remarquable d'espaces de {R}iemann},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {54},
  pages   = {214--264},
  year    = {1926}
}

@book{Besse1987Einstein,
  author    = {Besse, Arthur L.},
  title     = {Einstein Manifolds},
  publisher = {Springer-Verlag},
  year      = {1987},
  note      = {Ch. 7 (homogeneous Einstein metrics; naturally reductive spaces)}
}

Prerequisites

03.02.37
03.05.13
03.05.15
07.04.07
13.02.02

Tier anchors

beginner: Helgason *Differential Geometry, Lie Groups, and Symmetric Spaces* Ch. IV §1 (informal); Nomizu 1954 (the picture: connections that look the same at every point)
intermediate: Kobayashi-Nomizu *Foundations of Differential Geometry Vol. II* Ch. X §2 (invariant connections), §2 Thm 2.1 (the Nomizu bijection); Helgason Ch. IV §1-§2
master: Kobayashi-Nomizu Vol. II Ch. X §2-§4 (the canonical connection, parallel curvature and torsion, completeness); Nomizu 1954 (Amer. J. Math. 76, 33-65); Helgason Ch. IV §§1-3

References

Nomizu — Invariant affine connections on homogeneous spaces (1954) · Amer. J. Math. 76, pp. 33--65 (Theorems 8.1, 10.1; the canonical connection)
Kobayashi-Nomizu — Foundations of Differential Geometry, Vol. II (1969) · Ch. X §2 Thm 2.1 (the bijection); §2 Thm 2.10 (canonical connection); §2 Cor 2.5
Helgason — Differential Geometry, Lie Groups, and Symmetric Spaces (1978) · Ch. IV §1-§3 (invariant connections, the canonical connection, affine symmetric spaces)

Estimated time

beginner: 18m
intermediate: 46m
master: 82m