By Arthur L. Besse (auth.)

ISBN-10: 3642618766

ISBN-13: 9783642618765

ISBN-10: 3642618782

ISBN-13: 9783642618789

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Additional info for Manifolds all of whose Geodesics are Closed

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Basic Facts about the Geodesic Flow Using the first Bianchi identity and the above equality one also gets This follows easily from the Levi-Civita property by direct computation. Therefore, one often considers Rg as an element of ffz 1\ 2 T M. Let n be any two-plane and {v 1, v z } an orthonormal basis of n. We define ag(n), the sectional curvature of n, by One easily checks that ag does indeed depend only on n and not on the orthonormal basis {v 1 , vz } chosen. J. 87. In this paragraph a Riemannian metric g is fixed.

106Proposition. Let Y be a (normally parametrized) geodesic in M. 107 Proof We shall use standard arguments in Riemannian submersions. Let y be a geodesic which minimizes the energy of curves between mo and mi defined on [0,1]. We pick the point y(m o) in T,noM. The horizontallift on the curve y is precisely y since y is a geodesic. Consider any other curve c from y(mo) to Y(m 1 ). Since y minimizes the energy downstairs from mo to mi' we have IE9(PMOC) ~IEg(Y). On the other hand since g 1 ~ PZt(g) by construction, IEg,(C)~lEpt-(g)(C)=1E9(PMOC) ; we have equality if and only if c is horizontal, so that IEg,(y) =lEg(y).

Let y be a geodesic which minimizes the energy of curves between mo and mi defined on [0,1]. We pick the point y(m o) in T,noM. The horizontallift on the curve y is precisely y since y is a geodesic. Consider any other curve c from y(mo) to Y(m 1 ). Since y minimizes the energy downstairs from mo to mi' we have IE9(PMOC) ~IEg(Y). On the other hand since g 1 ~ PZt(g) by construction, IEg,(C)~lEpt-(g)(C)=1E9(PMOC) ; we have equality if and only if c is horizontal, so that IEg,(y) =lEg(y). Therefore IEg,(c)~lEg,(Y).