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By Freeadman M.H., Teichner P.

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F(m) If F takes on i t s minimum a t a p a t h a r e a l l defined, then c l e a r l y oo mo, and i f the i s a c r i t i c a l path. CALCULUS OF VARIATIONS 111. 70 THE FNERGY OF A PATH $12. 71 takes on i t s m i n i m u m d2 p r e c i s e l y on t h e s e t o f m i n i m a l geodesics from p t o q. Suppose now that tor v M i s a Riemannian manifold. TMP w i l l be denoted by E energy of from a t o b w We w i l l write E We w i l l now see which p a t h s The Energy of a Path. $12. \\v(\ (where 0 = 5a < v,v >*.

An, al , along = p. Since t h e W i ( l ) independent, no n o n - t r i v i a l l i n e a r combination of t h e Wi Since which vanish a t n n xi . constructed as above, provide vanishing a t yV, n Jacobi f i e l d s = exp,(Xi) are can vanish a t i s t h e dimension of the space of Jacobi f i e l d s along y V , p, c l e a r l y no n o n - t r i v i a l Jacobi f i e l d along a t b o t h p and exp v. yv vanishes T h i s completes t h e proof. 2. L e t p E M. Then f o r almost a l l p i s not conjugate t o q along any geodesic.

Tk) . To shorten t h e n o t a t i o n , l e t us denote this manifold Int n ( t o , t l , . ,t k ) c of broken geodesics by . 1. Let M be a complete Riemannian manifold; and l e t c be a f i x e d p o s i t i v e number such t h a t n c # 0 . , , tk) of [0,1 1 t h e s e t I n t n ( t o , t l , .. ,tk) can be given t h e s t r u c t u r e of a smooth f i n i t e dimensional manifold i n a natural way. B. B of t h e energy f u n c t i o n E : R +R. 111. CALCULUS OF VARIATIONS $1 THEOREM 1 6 . 2 .

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4-Manifold topology II: Dwyers filtration and surgery kernels by Freeadman M.H., Teichner P.


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