M ICHIHIKO K IKKAWA (Received: December 25, 1998) Abstract. Two kinds of canonical connections have been introduced by the author for homogeneous (left) Lie loops, one of which is the canonical con- nection of the reductive homogeneous spaces induced from the homogeneous (left) Lie loops and the other is the canonical connection of the homogeneous systems associated with left Lie loops. In this paper, it is shown that they are coincident with each other. Moreover, the concept of geodesic reductive homogeneous spaces are introduced. It is shown that the reductive homoge- neous space induced from a homogeneous left Lie loop is geodesic if and only if the left Lie loop is geodesic.
P RELIMINARIES
1.
For any linearly connected manifold with any fixed point e , the local binary operation is introduced by μ(x,y) = Exp x ◦ τε,x ◦ Exp − 1 (y) in some normal neighbourhood of e . Here, τ e,x denotes the parallel displacement of tangent vectors along the geodesic arc joining e to x . It has been proved that this local multiplication forms a local loop, called geodesic local loop at e  ([3], also cf. [1], [14]). Denote by L x the left translation by x . Non-associativity of μ is indicated by left inner maps L x,y := L − l − l − l y . Motivated by the concept of geodesic local loops on affinely connected manifolds, the algebraic concept of homogeneous loops and the concept of homogeneous Lie loops on manifolds have been introduced by the author in 1975 [4]. Let ( G , μ ) be an algebraic binary operation on a set G with the unit e such that all left translations L x : G → G ; L x y := μ(x,y), x ∈ G , are bijections of G . If the left translation L x satisfies ( L x )− 1 = L x − 1 for x− 1 = L x − 1 e, x ∈ G , it is called a left loop with the left inverse property . The left loop μ with the left inverse property is said to be homogeneous if all left inner maps L x,y are automorphisms of μ .  The subgroup of the automorphism group Aut( μ ) generated by all left inner maps is called the left inner mapping group of ( G , μ ). Moreover, if all right translations of the left loop are bijection, it is called a homogeneous loop.  Homogeneous (left) Lie loop is a homogeneous (left) loop defined on a differentiable manifold whose multiplication μ is differentiable.
2. HOMOGENEOUS LEFT LIE LOOPS AS REDUCTIVE HOMOGENEOUS SPACES
For any homogeneous left loop ( G, μ ), the concept of semi-direct product A = G × K of G by a group K is introduced, where K is a subgroup of Aut(G, μ ) containing the left inner mapping group of ( G, μ ). That is, for any ( x, α ) and (y, β) in A, define their product by; ( x, α)(y, β) := ( μ ( x, αy), L x,αy ◦ α ◦ β ). Then, A forms a group which is called the semi-direct product of G and K . Let ( G , μ ) be a homogeneous left Lie loop , K the closure of the left inner mapping group in the (differentiable) automorphism group of ( G, μ ). Then, the Lie group A of semi-direct product of G and K is called the enveloping Lie group of ( G, μ ). By using this, it is shown that any homogeneous left Lie loop is regarded as a reductive homogeneous space G = A/K . In fact, since any element α = ( e,α ) is an automorphism of ( G,μ ), it is easy to check the following relations:
for any x ∈ G and α ∈ K , which show that the submanifold G = ( G , id) of A is invariant under any inner automorphism of A by any element ( e, α) ∈ { e } × K = K . Settle on A/K the canonical connection of the reductive homogeneous space of Nomizu [13]. Then the homogeneous left Lie loop G is said to be geodesic if the multiplication μ is coincident with the geodesic local loop of the canonical connection, in some neighbourhood of the unit e. Especially, any Lie group ( G, μ ) is a geodesic homogeneous left Lie loop whose left inner maps are equal to the identity map. In this case, the canonical con- nection is reduced to the (-)-connection of E. Cartan. On the other hand, assume that a homogeneous left Lie loop ( G, μ ) satisfies the relation;
This relation means that the inversion J : G → G; J(x) := x-1, is an auto- morphism of ( G, μ ). Then, it is shown that the reductive homogeneous space G = A/K is reduced to an affine symmetric space. So, it is called a symmetric loop (cf. [4]). It has been shown that any symmetric loop is geodesic ([4]).
3. C ANONICAL CONNECTIONS
For any left Lie loop ( G , μ ) with the left inverse property, let η : G×G×G → G
be the homogeneous system associated with ( G , μ ) (cf. [7]). It has been shown in [10] that the canonical connection ∇ of the left Lie loop ( G, μ ) is given explicitly by the formula:
for any vector fields X, Y on G. Here, we denote
and, if u,v,w are considered as independent coordinate variables in the 3n- variable functions η( u, ν, w) k ,  k = 1, ･ ･ ･ , n , on the manifold G of dimension n , we denote
in some coordinate neighborhood of the unit e. Especially, this is valid for homogeneous left Lie loops. In this case, the canon- ical connection given above is coincident with the canonical connection of reduc- tive homogeneous space A/K for the semi-direct product A = G × K mentioned in the previous section. That is; Theorem 3.1. Let ( G , μ ) be a homogeneous left Lie loop, A = G × K the semi- direct product of G by the closure K  of the left inner mapping group of (G, μ). Let ∇ be the canonical connection of ( G , μ ) defined by the formula (3.1). Then, the connection ∇ is an invariant connection of the reductive homogeneous space G = A/K which is coincident with the canonical connection of A/K. Proof. Let α be any (differentiable) automorphism of ( G, μ ). For any two vector fields X, Y on G, we can see dα(X x Y − η(x,X x,Y x )) = (dαX)αxdαY − η(αx,(dαX)αx,(dαY)αx). So the connection ∇ is invariant under the group K . Let t ( x,α) : G → G, ( x,α) ∈ A denote the action of A on the manifold G .  By regarding the semi-direct product as A = ( G , id) × ( e, K ), the action of ( x, id) on y = ( y, K ) ∈ G = πA is expressed by t( x,id )( y, K ) = ( μ ( x, y ), K ), for any x, y ∈ G . This means that the element ( x, id) of A acts on G as the left translation L x . For each tangent vector X ∈覧= T e ( G ) at the origin e = π ( e, K ), consider a vector field X * on some neighborhood U of e given by
By the original definition of K.Nomizu [13], the canonical connection ∇ * (of 2nd kind) of a reductive homogeneous space is an invariant connection satisfying (3.2)
for any X,Y E &. So, it is sufficient to show that the connection given by (3.1) satisfies the relation (3.2). Since μ( x, y ) = η( e, x, y ), the vector field X* can be described in terms of the homogeneous system η as X * = η(e, x, X). Then, we have
On the other hand, the equalities
hold so that we have
4. G EODESIC REDUCTIVE HOMOGENEOUS SPACES Let G = G/K be a reductive homogeneopus space of a Lie group G = (G, ji) by a closed subgroup K , with the projection π :  G → G and the decomposition of the Lie algebra adK ৫ C ৫, & = & ⊕ &, where & is a subspace of & which is regarded as the tangent space to G at the origin e = π( K ), and R is the Lie algebra of K . Let exp be the exponential map of the Lie algebra & into G . Consider the geodsic local loop ( U, μ ) on a normal neighborhood U of e , with respect to the canonical connection of the re- ductive homogeneous space. We introduce the new concept of geodesic reductive homogeneous spaces: Definition 4.1. The reductive homogeneous space G = G/K will be said to be geodesic if the diagram
commutes for some normal neighborhood U of the origin e , where Ŭ = π− 1 ( U ) f exp( 6 ). Theorem 4.1. Let (G, μ) be a homogeneous left Lie loop on a connected differ- entiable manifold G, A = G × K  the enveloping Lie group by the closure K  of the left inner mapping group in Aut(G, μ).  Then, (G, μ) is geodesic if and only if the reductive homogeneous space A/K is geodesic. Proof. Assume that the homogeneous left Lie loop ( G, μ ) is geodesic. For any tangent vector X ∈ T ε ( G ) at the unit e , the geodesic curve x ( t ), t ∈ R , tangent to X at e = x (0) is a one-parameter subgroup of the left loop ( G, μ ). Moreover, it satisfies the relation L x (t),x (s) = id for any t,s ∈ R (cf. [4] pp. 163 −168) . Consider a curve given by ñ(t) := ( x (t), id), t ∈ R, in the Lie group A = G × K. Then, it is a one-parameter subgroup of A. In fact we have ( x ( t ), id)( x ( s ), id) = ( μ ( x ( t ), x ( s )) = ( x ( t + s ), id), t, s ∈ R. For some normal neighborhood U at e, set Ü := π-1(U) ∩ exp(¢) in the Lie group A whose Lie algebra is assumed to be decomposed into λ =覧⊕ R by the tangent Lie triple algebra & of ( G, μ ) and the Lie algebra R of K . Then, any two elements ¥, ÿ ∈ U are given by some one-parameter subgroups ¥(t), ÿ(s), t, s ∈ R , in A as ñ = ñ(t o ), ñ = ñ(s o ) for some t o , s 0 ∈ R. The group multiplication of ñ and ỹ is given by ji( ž, ỹ) = ( x ( t 0 ), id)( y ( s 0 ), id) = ( μ ( x ( t 0 ), y ( s 0 )), L x ( t 0 ),y ( s 0 ) . Thus we get
that is, the diagram (4.1) commutes so that the reductive homogeneos space G = A/K is geodesic. Conversely, assume that the reductive homogeneous space A/K is geodesic. For some normal neighborhood U at e , any element ī = ( x,α) ∈ Ŭ = π-1 ( U ) ∩ exp( & ) can be regarded as an element̃x =̃x ( t 0 ) of some one-parameter subgroup ž(t),t ∈ R, in the Lie group A. Theorem 3.1 implies that the one-parameter subgroup ī ( t ) acts on G by some parallel displacement τ e,x(t) : T e ( G ) → T z(t) ( G ) of tangent vectors along the geodesic curve x ( t ) = π x ( t ) passing through e = x(0). So, if we set x(t) = (x(t), α(t)), we get
Te,x(t) = dLx(t) ◦ dα(t),t ∈ R. On the other hand, by the assumption on the reductive homogeneous space A/K, α(t) acts on U as the identity transformation. In fact, for any y ∈ U, set π−1(y) = (y,β) ∈ Ü. Then we get and
These must coincide with each other so that α(t)y = y for y ∈ U. Hence, the parallel displacement τe,x(t) of tangent vectors along the geodesic curve x(t) comes from the left translation L x (t) of the left loop ( G, μ ). Since L x (t) Y = η(e, x (t), y), the left translation L x ( t ) is an affine transformation of the canonical connection. Thus we see that L x (t) brings any geodesic y (s) passing through e = y (0) to the geodesic z(s) = μ(x(t),y(s)) through x(t) = z(0). This means that the multiplication μ is coincident with that of the geodesic local loop on U . ●
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