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By George D. Birkhoff, Ralph Beatley

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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.

AI (M) denotes the inyerse of ti. In particular, if fEe (M), then the vector field (dr)" is called the gradient off co frame, where and it is denoted by grad{/) (see also §2 of Chapter 2). This vector field is characterized, therefore, by the relation g(grad(J),y) = df(Y) =r(f). Y E ::t(M) (see also §2 of Chapter 2). If L: ::t(M) --? ::t(M) is a smooth (= C) for each tensor field of type (1,1), then the map 1:::t2 (M) --? C (M) given by i(x. 2) on M. Conversely. if F:::t) (M) --? 2) on M, then there exists a unique smooth tensor field P:::t (M) --?

1) (i 0 (/) )(/),-1 (f)) = (/), [(/),-1 (X )(10 (/) )(/),-1 (f) + (i c (/))9 <1>"11'1(/),-1 (f)] = (/). [( X (1)0 (/) )(/). I (};)] + (/). [(I, (/))'! <1>. i )(/), I (f l] = x(i)f + 1(/), (v <1>. 'II')(/)' I (f·)) for each 1 E ex (Ail) and X,Y E ::r (Ail ). D. g) be a Riemannian manifold. 'V the Levi-Civita connection. (U,x I , .. \1. \ YI f = L f--[f--. k _I IR. are defined by: 9 r. ~=f--r ~ .... k ~ /k .... (~X The r;k ex I 1 ex I· are the so-called second Christoffel symbols and these can be expressed only in terms of the first order derivatives of the metric tensor field: 3.

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