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Thus the rank of the system (that is the rank of 02) is 8, or equivalently: dim K — 1 . This example enables us to understand how K and r can be found : One writes the system which defines Im Ok+i ■ Then: - T is defined by the relations between the equations system; - dim K is the number of these relations. 10 Let us consider a manifold M endowed with a linear connec­ tion V. —► VX where VX : X(M) Y ► X(M) ►—>■ VYX. We have the following diagram: S2T*ig>T g2(V) > T'®(T'®T) > i- ► Cokera 2 (V) >0 i- R2 ► J2T Pl(V) Ri ► JiT Po(V) ) Ji(r*®T) (T'giT) As we have already seen (cf.

On the other hand S does not depend on the choice of S, because if S' is another spray J(S-S') = C-C = 0; then S - S' is vertical so h(S - S') - 0. Locally: where the T^x,])) are the coefficients of the connection. 3 The paths of the spray associated to the T are the geodesies ofT. connection Indeed, for any curve 7 on M, one has : «/7» = C7< . e. S7< = h-y" = 7 " — vy". The property follows from this equality. 4 (cf. [Gr]). Let S be a spray on M. Then T := [J,S] is a connection. The spray associated to [J,S] is S + ^S*, where S* is the deflection of S.

1-6) Let consider the map: r : T" A 2 T* —> A 3 T* defined by T(C)(X, Y, Z) := C(X, y, Z) + C(Y, Z, X) + C{Z, X, Y). 6) is exact. In fact, we have that r o a2 is zero, so Im