~ Ns let c(t) , t c J , be a a) f(x(t)) b) ( f ( x ( t ) ) , Y t) lel) in vector t f,TM and isometric of (rasp. -parallel to be an a f f i n e p a r a m e t e r There exists c in M (i) ing C~ for a g e o d e s i c geodesic (We r e q u i r e be a p a r a l l e l for is a F r e n e t (b) TM x(t) (rasp. Tx(t)M £ Vi(t) , i ] ~Nf(x(t))M while if c is in E Proof: The Frenet Let even , , i odd , even . VtX t + ~ ( X t , X t) = ~(Xt,X t) since x(t) ~(t) of c is constant. t ~ = 0 ; (~3tx) (t) = -f,VtA u(xt,xt )X t - e(A e(Xt,Xt )x t,x t) = -e(Ae(xt,xt)Xt,Xt) In particular, for j (~Jx) (t) odd.

2) cancel, so that *V(cIII)(Z,Z,Z) = ~Z H. 3) 53 The proof of the theorem now follows immediately. We will now obtain a local expression for ~I 1 using moving frames. This will enable us to compare our tensor fields with those constructed by Jensen and Rigoli. ,e n} be a local orthonormal framing of ~*TW such that {el,e 2} is a positively oriented Jensen and Rigoli call this a "first basis of the tangent space to M. Let {~I,~2} order fr~ne field". __ ml + i~2. be the basis dual to {e l,e 2} and let Then fl I = ~+2, where f is the CNM-valued function given by { - 4| ~i I (el - ie2,e I - ie2) - 4!