- £-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 357
the proof of the Riemannian index lemma. We will give the details of the
proof elsewhere.
LEMMA 7.99 (£-cut locus).
(i) £ Cut(p,0) (r) = £ Cutzp,O) (r) u £ Cut(p,O) (r) and is closed.
(ii) £ Cut~,o) ( r) has measure 0.
·(iii) £ Cut(p,O) (r) is closed and has measure 0.The proof of the above lemma depends on an index lemma for £-length.
Define the £-index form £ I(Y, W) by
r
!V'y\i'wR+ (R(Y,X) W,X) ]
(7.134) £I(Y,W)~ 1:b VT +(V'xY,V'xW)-(V'yRc)(W,X) dT.
- (V'w Re) (Y, X) + (V'x Re) (Y, W)
Note that the second variation of £-length (7.62) is related to the £-index
form by
(7.135) (o}£) (1) = 2VT (V'yY,X)I~: + 2£I(V, V).
On the other hand, the £-index form is related to the £-Jacobi equation by
£I(Y, W) =VT (V'xY, W)I~:
1
(^7) b [! V'x (V'xY) - Rm (X, Y) X - !V'y (V'R) )]
- Ta VT \ +2 (V'y Re) (X) + 2Rc (V'xY) + d- 7 V'xY 'W dT.
This can be proved by integrating by parts on the term ft (V' x Y, V' x W)
in (7.134).
LEMMA 7.100 (£-index lemma). Let r be an £-geodesic from (p, Ta) to
(q, Tb) such that there are no points £-conjugate to (p, Ta) along f. For any
piecewise smooth vector field W along r with W (Ta) = 0, let Y be the unique
£-Jacobi field such that Y(Ta) = W(Ta) = 0 and Y(Tb) = W(Tb)· Then
£I(Y, Y) :S £I(W, W)and the equality holds if only if Y = W. Here we have used the obvious
generalization of the definition of £-conjugate point with (p, 0) replaced by
(p, Ta).Lemma 7.99 implies the following.COROLLARY 7.101 (Differentiability of L away from £-cut locus). Givenr E (0, T) and q EM, suppose that there is only one minimal £-geodesic r
joining (p, 0) and (q, r) and suppose that (q, r) is not an £-conjugate point
of (p, 0), i.e., (q, r) is not an £-cut locus point. Then the L-distance L(-, ·)
and reduced distance£ are C^2 -differentiable at (q, r).