By Arthur L. Besse (auth.)

ISBN-10: 3642618766

ISBN-13: 9783642618765

ISBN-10: 3642618782

ISBN-13: 9783642618789

Show description

Read Online or Download Manifolds all of whose Geodesics are Closed PDF

Best differential geometry books

New PDF release: The topology of fibre bundles

Fibre bundles, now a vital part of differential geometry, also are of serious value in glossy physics - similar to in gauge thought. This ebook, a succinct creation to the topic via renown mathematician Norman Steenrod, was once the 1st to offer the topic systematically. It starts with a common advent to bundles, together with such issues as differentiable manifolds and overlaying areas.

Differential geometry and complex analysis: a volume by I. Chavel, H.M. Farkas PDF

Chavel I. , Farkas H. M. (eds. ) Differential geometry and complicated research (Springer, 1985)(ISBN 354013543X)(236s)

Download PDF by Leon Simon: Theorems on regularity and singularity of energy minimizing

The purpose of those lecture notes is to provide an basically self-contained advent to the elemental regularity thought for strength minimizing maps, together with contemporary advancements in regards to the constitution of the singular set and asymptotics on method of the singular set. really expert wisdom in partial differential equations or the geometric calculus of adaptations is no longer required.

Additional info for Manifolds all of whose Geodesics are Closed

Sample text

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).

Download PDF sample

Manifolds all of whose Geodesics are Closed by Arthur L. Besse (auth.)


by James
4.2

Rated 4.32 of 5 – based on 45 votes