Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over C x T admit no complete nonnegatively curved metrics.

[1]  Burkhard Wilking.  Manifolds with positive sectional curvature almost everywhere , 2002 .

[2]  Burkhard Wilking.  On fundamental groups of manifolds of nonnegative curvature , 2000 .

[3]  K. Grove,et al.  Curvature and symmetry of Milnor spheres , 2000, math/0007198.

[4]  I. Belegradek,et al.  Finiteness theorems for nonnegatively curved vector bundles , 2000, math/0002028.

[5]  I. Belegradek,et al.  Topological obstructions to nonnegative curvature , 2000, math/0001125.

[6]  Luis Guijarro,et al.  The metric projection onto the soul , 1999 .

[7]  J. Rutter.  Spaces of Homotopy Self-Equivalences - A Survey , 1997 .

[8]  A. L. Onishchik.  Topology of Transitive Transformation Groups , 1997 .

[9]  J. Rutter.  Spaces of Homotopy Self-Equivalences , 1997 .

[10]  J. Oprea,et al.  Symplectic manifolds with no Kähler structure , 1997 .

[11]  Y. Bazaikin.  On a certain family of closed 13-dimensional Riemannian manifolds of positive curvature , 1996 .

[12]  Dagang Yang.  On complete metrics of nonnegative curvature on $2$-plane bundles. , 1995 .

[13]  M. Özaydin,et al.  Vector bundles with no soul , 1994 .

[14]  Pierre-Paul Grivel.  Algèbres de Lie de dérivations de certaines algèbres pures , 1994 .

[15]  W. Singhof.  On the topology of double coset manifolds , 1993 .

[16]  J. Eschenburg.  Cohomology of biquotients , 1992 .

[17]  J. Eschenburg.  Inhomogeneous spaces of positive curvature , 1992 .

[18]  S. Stolz.  Simply connected manifolds of positive scalar curvature , 1990 .

[19]  G. Lupton.  Note on a conjecture of Stephen Halperin's , 1990 .

[20]  Mckenzie Y. Wang,et al.  Einstein metrics on principal torus bundles , 1990 .

[21]  Yves Félix.  La dichotomie elliptique-hyperbolique en homotopie rationnelle , 1989 .

[22]  J. Mccleary,et al.  ON THE FREE LOOP SPACE OF HOMOGENEOUS SPACES , 1987 .

[23]  H. Shiga,et al.  Rational fibrations homogeneous spaces with positive Euler characteristics and Jacobians , 1987 .

[24]  W. Meier.  Some topological properties of Kähler manifolds and homogeneous spaces , 1983 .

[25]  K. Grove,et al.  Contributions of rational homotopy theory to global problems in geometry , 1982 .

[26]  J. Eschenburg.  New examples of manifolds with strictly positive curvature , 1982 .

[27]  W. Meier.  Rational universal fibrations and flag manifolds , 1982 .

[28]  Jean-Claude Thomas.  Rational homotopy of Serre fibrations , 1981 .

[29]  T. J. Miller,et al.  Formal and coformal spaces , 1978 .

[30]  A. Rigas.  Geodesic spheres as generators of the homotopy groups of ${\rm O}$, $B{\rm O}$ , 1978 .

[31]  Dennis Sullivan,et al.  Infinitesimal computations in topology , 1977 .

[32]  S. Halperin.  Finiteness in the minimal models of Sullivan , 1977 .

[33]  W. Greub,et al.  Connections, curvature and cohomology , 1976 .

[34]  W. Greub,et al.  Cohomology of principal bundles and homogeneous spaces , 1976 .

[35]  J. Cheeger.  Some examples of manifolds of nonnegative curvature , 1973 .

[36]  N. Wallach.  Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature , 1972 .

[37]  Charles Terence Clegg Wall,et al.  Surgery on compact manifolds , 1970 .

[38]  L. Siebenmann.  On detecting open collars , 1969 .

[39]  Detlef Gromoll,et al.  The structure of complete manifolds of nonnegative curvature , 1968 .

[40]  A. Haefliger.  Plongements différentiables de variétés dans variétés , 1962 .