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Download Lectures on Closed Geodesics by W. Klingenberg PDF

By W. Klingenberg

The query of life of c10sed geodesics on a Riemannian manifold and the houses of the corresponding periodic orbits within the geodesic circulate has been the article of in depth investigations because the starting of worldwide differential geo­ metry over the last century. the best case happens for c10sed surfaces of unfavourable curvature. the following, the elemental team is massive and, as proven by means of Hadamard [Had] in 1898, each non-null homotopic c10sed curve should be deformed right into a c10sed curve having minimallength in its unfastened homotopy c1ass. This minimum curve is, as much as the parameterization, uniquely made up our minds and represents a c10sed geodesic. The query of life of a c10sed geodesic on a easily attached c10sed floor is way more challenging. As mentioned through Poincare [po 1] in 1905, this challenge has a lot in universal with the matter ofthe lifestyles of periodic orbits within the constrained 3 physique challenge. Poincare [l.c.] defined an evidence that on an analytic convex floor which doesn't range an excessive amount of from the normal sphere there continually exists at the least one c10sed geodesic of elliptic kind, i. e., the corres­ ponding periodic orbit within the geodesic movement is infinitesimally stable.

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Remark. The additive structure of H * (G(2, n -1) ) and H* (G(2, n -1) ) was determined by Ehresmann [Eh]. The multiplicative structure of H* (G(2, n-l» was determined by Chem [Ch]. An alternative way of describing the cohomology ring of G(2, n-l) is as follows, cf. Borel [Bor 1]: Denote by S(Xl'"'' xp) the algebra of symmetric polynomials in X l " ' " Xp' and by S+(x l " . " xp) the subalgebra generated by the non-constant polynomials. Then H* (G(2, n -1») can be written as S(u l , u Z)®S(U3,' ..

Corresponding to every conjugacy class of. 1 of the fundamental group 11:1M of M we have a connected component A' of A=AM which does not contain the set AO ofpoint curves. Claim. The function E assumes its infimum K' on A', and E- 1(K')nA' consists of closed geodesics. Proof Recall that the set A' is a ,p-family. Its critical value K' is positive since a curve c with E(c), and hence L(c) (the length of c), sufficiently small will be null homotopic. If cEE- 1(K')nA' were not a critical point of E, then IIgrad E(c)111 >0 and E(,psC)

Co is called the underlying prime curve of c. We define the space of unparameterized oriented closed curves on M, nM, as the quotient space of AM with respect to the S-action X-: Let fi: AM-+nM=AM/ x-S be the quotient map. e. a set Ben will be open if and only if the counterimage of B under is open. Clearly, E: nM->JR. is a continuous function. We call CEn critical if C is the image under fi of a critical point of A. Another way of describing the space would be to say that it consists of the orbits of the S-action X-.

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