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Contributions of rational homotopy theory to global problems in geometry. (English) Zbl 0508.55013

MSC:
55P62 Rational homotopy theory
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C22 Geodesics in global differential geometry
57R19 Algebraic topology on manifolds and differential topology
53C20 Global Riemannian geometry, including pinching
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References:
[1] I. K. Babenko, On analytic properties of the Poincaré series of loop spaces,Math. Zametki,27 (1980), 751–765.
[2] J. Barge, Structures différentiables sur les types d’homotopie rationnelle simplement connexes,Ann. Scient. Éc. Norm. Sup.,9 (1976), 469–501. · Zbl 0348.57016
[3] M. Berger, R. Bott, Sur les variétés à courbure strictement positive,Topology,1 (1962), 301–311. · Zbl 0112.13604 · doi:10.1016/0040-9383(62)90017-4
[4] Y. Felix, S. Halperin, Rational Lusternik-Schnirelmann category and its applications,Trans. Amer. Math. Soc.,273 (1982), 1–37. · Zbl 0508.55004 · doi:10.2307/1999190
[5] Y. Felix, S. Halperin, J. C. Thomas, The homotopy Lie algebra for finite complexes,Publ. Math. I.H.E.S., ce volume, 179–202. · Zbl 0504.55005
[6] Y. Felix, J. C. Thomas, The radius of convergence of Poincaré series of loop spaces,Invent. math.,68 (1982), 257–274. · Zbl 0488.55009 · doi:10.1007/BF01394059
[7] J. B. Friedlander, S. Halperin, Rational homotopy groups of certain spaces,Invent. math.,53 (1979), 117–133. · Zbl 0408.55010 · doi:10.1007/BF01390029
[8] M. Gromov, Curvature, diameter and Betti numbers,Comment. Math. Helv.,56 (1981), 179–195. · Zbl 0467.53021 · doi:10.1007/BF02566208
[9] K. Grove, Condition (C) for the energy integral on certain path-spaces and applications to the theory of geodesics,J. Differential Geometry,8 (1973), 207–223. · Zbl 0277.58004
[10] K. Grove, Isometry-invariant Geodesics,Topology,13 (1974), 281–292. · Zbl 0289.58007 · doi:10.1016/0040-9383(74)90021-4
[11] K. Grove, S. Halperin, M. Vigué-Poirrier, The rational homotopy theory of certain path-spaces with applications to geodesics,Acta math.,140 (1978), 277–303. · Zbl 0421.58007 · doi:10.1007/BF02392310
[12] K. Grove, M. Tanaka, On the number of invariant closed geodesics,Acta math.,140 (1978), 33–48. · Zbl 0375.58010 · doi:10.1007/BF02392302
[13] S. Halperin, Finiteness in the minimal model of Sullivan,Trans. Amer. Math. Soc.,230 (1977), 173–199. · Zbl 0364.55014 · doi:10.1090/S0002-9947-1977-0461508-8
[14] S. Halperin, Spaces whose rational homotopy and \(\psi\)-homotopy are both finite dimensional, to appear. · Zbl 0546.55015
[15] H. Hernández-Andrade, A class of compact manifolds with positive Ricci curvature,Proc. symp. pure math. A.M.S., XXVII (1975), 73–87.
[16] J. Milnor,Singular points of complex hypersurfaces, Annals of math. studies,61 (1968), Princeton. · Zbl 0184.48405
[17] S. B. Myers, N. Steenrod, The group of isometries of a Riemannian manifold,Ann. of Math.,40 (1939), 400–416. · Zbl 0021.06303 · doi:10.2307/1968928
[18] D. Quillen, Rational homotopy theory,Ann. of Math.,90 (1969), 205–295. · Zbl 0191.53702 · doi:10.2307/1970725
[19] D. Sullivan, Infinitesimal computations in topology,Publ. Math. I.H.E.S.,47 (1978), 269–331. · Zbl 0374.57002
[20] M. Tanaka, On the existence of infinitely many isometry-invariant geodesics,J. Differential Geometry,17 (1982), 171–184. · Zbl 0499.53041
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