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6. 7 Let M carry a Riemannian metric g = gijdxi ® dxi. A vector field X on M is cailed Killing field or infinitesimal isometry if Lx(g) = 0. 6. 7 A vector field X on a Riemannian manifold M is a Killing field if and only if the local1-parameter group generated by X consists of local isometrics. 21) :t(1/J;g)Jt=0 = 0. 22) Since this holds for every point of M, we obtain 1/J; g =g for all t E I. Therefore, the diffeomorphisms 1/Jt are isometries. 21). 8 The Killing fields of a Riemannian manifold constitute a Lie algebra.

6. 4. 7 Let M be a compact Riemannian manifold. Then for any p E M, the exponential map expp is defined on all of TpM, and any geodesic may be extended indefinitely in each direction. Proof. For v E TpM, let A:= {t E JR+ : Cv is defined on [-t, t]}, where Cv is, as usual, the geodesic with Cv(O) = p, cv(O) = v. It follows from cv(-t) = c_v(t) that ev may also be defined for negative t, at the moment at least for those with sufficiently small absolute value. 2 implies A ::j:. 0. The compactness of M implies the closedness of A.

The Jacobi identity follows by direct computation. 5 A Lie algebra (over IR) is a real vector space V equipped with a bilinear map[·,·] : V x V ~ V, the Lie bracket, satisfying: (i) [X, X]= 0 for all X E V. (ii) [X, [Y, Z]] + [Y, [Z, X]]+ [Z, [X, Y]] = 0 for all X, Y, Z E V. 2 The space of vector fields on M, equipped with the Lie bracket, is a Lie algebra. 6 Let '1/J : M M. 12) Thus, '1/J* induces a Lie algebra lsomorphism. Proof. 3. 0 We now want to investigate how one might differentiate tensor fields.