THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

We also have:D S,T −t dπ ˜// 0,te i = V −1dπ ˜// 0,t e i((∗d(dπ ˜// 0,te i )) v T −t)= V −1dπ ˜// 0,t e i((ddπd ev ei (∗d ˜// 0,t)) v T −t)= V −1dπ ˜// 0,t e i(ddπ(d ev ei (∗d ˜// 0,t ))ṽ)= 0.Where we have used in the last equality the fact that ˜// 0,tis the ˜∇ horizontal liftof Y t . The third one may be seen for curves, it comes from the definition of ˜∇.Following computations similar to one in the first section, we have by (1.5):.∗d((// T 0,t) −1 dπ ˜// 0,t) i,j = ∂ ∂ tg(T − t)(dπ ˜// 0,te i , Ut T ẽ j ) dt+ 〈dπ ˜// 0,te i , D S,T −t Ut T ẽ j 〉 T −t= ∂ ∂ tg(T − t)(dπ ˜// 0,te i , Ut T ẽ j ) dt+ 〈dπ ˜// ∑0,te i , − 1 d ∂2 α=1 ∂ tg(T − t)(Ut T ẽ j , Ut T ẽ α )Ut T ẽ α 〉 T −t dt= ∂ ∂ tg(T − t)(dπ ˜// 0,te i , Ut T ẽ j ) dt∑− 1 d ∂2 α=1 ∂ tg(T − t)(Ut T ẽ j , Ut T ẽ α )〈dπ ˜// 0,te i , Ut T ẽ α 〉 T −t dt= 1 ∂2 ∂ tg(T − t)(dπ ˜// 0,te i , Ut T ẽ j ) dt.In the general case, and by previous identification:d((// T 0,t) −1 dπ ˜// 0,t)(0, e i ) = 1 2∑j∂∂ tg(T − t)(dπ ˜// 0,te i , U T t ẽ j )e j dt (4.3)= 1 2 (//T 0,t) −1 ( ∂ ∂ tg(T − t)) #T −t (dπ ˜// 0,t(0, e i )) dt.(4.4)Differentiating (4.2) along a geodesic curve beginning at (0, x 0 ) with velocity(v t , v) and using corollary 3.17 in [3] we get:˜// 0,td ( ˜//−10,t T Y t(v t , v) ) = − 1 2 ˜R(T Y t (v t , v), dY t (x 0 ))dY t (x 0 ),where ˜R is the curvature tensor.Let v ∈ T x M we write:T X t v := dπT Y t (0, v).In a more canonical way than theorem 3.2, we have the following proposition.5722