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[0, +00) the usual norm. In the case of( P" (IC). 16). for all II d (x,y) = J (lZ'-1 (x ),lZ'-1 (y)) for all x. yEP" (iC). e. d is just the Fubini-Study metric on P" (iC). ')\~·can). g) is given by the following formula: cosh(d(x y))= 1+ Ilx- ylll , 2x y" ' fl 2.

Gw,,)] acts freely on H n (resp. ") if and only if r is torsion-free. On the other hand, one can show that any simply-connected complete n-dimensional Riemannian manifold of constant sectional curvature + 1 (resp. 0, resp. -1) is isometric to (S", gra,,) [resp. ". gum). resp. (Hn, g)], n? 2. For reasons that will be clear later. let us point out, some geometrical objects that can be derived from the curvature tensor field. e. the tensor field of type (0,2) on M given by: (c1R)' (u, v) = trace[w H R, (u, w)v], where u, v, wE T,M and x EM.

One can define the covariant differential , 'VXE9i(M) of XE::t(M) by for any Y E ::t ( M) . 1 (Levi-Civita). On any Riemannian manifold eM,g) there exists a unique symmetric linear connection consistent with the metric tensor field g. e. Z)+ g(r. \\2). Z E ::t(M). Proof. First assume that there exists a linear connection V on M satisfying (iii) and (iv). Z) + g (Y. r)+ g(X. 'VzY). Adding the first two equalities. substracting the last one and using (iii) we obtain: 3. The Levi-Civita Connection 31 x (g(Y,Z))+ Y(g(Z,X)) - Z (g(X.