Page - 78 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 R ( ∂ ∂xk , ∂ ∂xl ) ∂ ∂xj =∑ i Rijkl ∂ ∂xi withRijkl= ∂Γilj ∂xk − ∂Γikj ∂xl +∑ m ( ΓmljΓ i km−ΓmkjΓilm ) (116) TheRicci tensorRicofD isgivenby: Ric(Y,Z)=Tr{X→R(X,Y)Z} (117) Rjk=Ric ( ∂ ∂xj , ∂ ∂xk ) =∑ i Rikij (118) In the following, we will consider a homogeneous space G/K endowed with a G-invariant flat connection D (homogeneous flat manifold) written (G/K, D). Koszul has proved a bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra ofG. Let (G, K) be the pair of connected Lie group G and its closedsubgroupK. Let g theLiealgebraofG andkbe theLie subalgebraof g corresponding toK. X∗ isdefinedas thevectorfieldonM=G/K inducedbythe1-parametergroupof transformation e−tX.WedenoteAX∗=LX∗−DX∗,withLX∗ theLiederivative. LetVbethe tangentspaceofG/Kato={K}andletconsider, the followingvaluesato: f(X)=AX∗,o (119) q(X)=X∗o (120) whereAX∗Y∗=−DY∗X∗ (whereD isa locallyflat linearconnection: its torsionandcurvature tensors vanish identically), then: f ([X,Y])= [f(X), f(Y)] (121) f(X)q(Y)− f(Y)q(X)= q([X,Y]) (122) whereker(k)= q, and (f,q)anaffinerepresentationof theLiealgebrag: ∀X∈ g, Xa=∑ i ( ∑ j f(X)jix i+q(X)i ) ∂ ∂xi (123) The1-parameter transformationgroupgeneratedbyXa isanaffinetransformationgroupofV, with linearpartsgivenby e−t.f(X) andtranslationvectorparts: ∞ ∑ n=1 (−t)n n! f(X)n−1q(X) (124) Theserelationsareprovedbyusing:⎧⎨⎩ AX∗Y ∗−AY∗X∗=[X∗,Y∗] [AX∗,AY∗]=A[X∗,Y]∗ withAX∗Y∗=−DY∗X∗ (125) basedontheproperty that theconnectionD is locallyflatandthere is local coordinatesystemsonM suchthatD ∂ ∂xi ∂ ∂xj =0withavanishingtorsionandcurvature: T(X,Y)=0⇒DXY−DYX=[X,Y] (126) R(X,Y)Z=0⇒DXDYZ−DYDXZ=D[X,Y]Z (127) deducedfromthefact thea locallyflat linearconnection(vanishingof torsionandcurvature). 78