Weakly complex homogeneous spaces

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.

[1]  Reinhard Racke,et al.  Elastic and electro-magnetic waves in infinite waveguides , 2008 .

[2]  Norbert Knarr,et al.  Polarities and planar collineations of Moufang planes , 2013 .

[3]  Ari Laptev,et al.  Geometrical Versions of improved Berezin-Li-Yau Inequalities , 2010, 1010.2683.

[4]  Wolfgang Kimmerle,et al.  On Torsion Subgroups in Integral Group Rings of Finite Groups , 2011 .

[5]  Markus Stroppel,et al.  Buttons, Holes and Loops of String: Lacing the Doily , 2012 .

[6]  J. Frank Adams,et al.  Lectures on Lie groups , 1969 .

[7]  J.dieudonne.  Treatise On Analysis Vol-ii , 1976 .

[8]  Rupert L. Frank,et al.  Eigenvalue estimates for Schrödinger operators on metric trees , 2007, 0710.5500.

[9]  T. Weidl,et al.  Improved Berezin-Li-Yau inequalities with a remainder term , 2007, 0711.4925.

[10]  Gil Solanes,et al.  Horo-tightness and total (absolute) curvatures in hyperbolic spaces , 2013 .

[11]  Markus Stroppel,et al.  Stabilizers of Subspaces under Similitudes of the Klein Quadric, and Automorphisms of Heisenberg Algebras , 2010, 1012.0502.

[12]  Andrei Moroianu,et al.  Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces , 2010, 1003.5172.

[13]  R. Hermann.  Compact homogeneous almost complex spaces of positive characteristic , 1956 .

[14]  Markus Stroppel,et al.  Polarities of Schellhammer Planes , 2011 .

[15]  J. Adams,et al.  Lectures on Exceptional Lie Groups , 1996 .

[16]  Barbara Kaltenbacher,et al.  On convergence of local averaging regression function estimates for the regularization of inverse problems , 2011 .

[17]  Thomas Merkle.  On semiclassical and universal inequalities for eigenvalues of quantum graphs , 2010 .

[18]  W. Singhof,et al.  Parallelizability of homogeneous spaces, II , 1982 .

[19]  Adam Krzyżak,et al.  Estimation of the essential supremum of a regression function , 2011 .

[20]  Wolfgang Kimmerle,et al.  Finite groups of units and their composition factors in the integral group rings of the groups PSL(2, q) , 2008, 0810.0186.

[21]  László Györfi,et al.  Strongly consistent nonparametric tests of conditional independence , 2012 .

[22]  M. Hamilton.  ITERATED FIBRE SUMS OF ALGEBRAIC LEFSCHETZ FIBRATIONS , 2012, 1209.2653.

[23]  Michael Kohler,et al.  Weakly universally consistent static forecasting of stationary and ergodic time series via local averaging and least squares estimates , 2013 .

[24]  J. Dieudonne.  Treatise on Analysis , 1969 .

[25]  A. Borel,et al.  Les sous-groupes fermés de rang maximum des groupes de Lie clos , 1949 .

[26]  Felix Effenberger,et al.  Stacked polytopes and tight triangulations of manifolds , 2009, J. Comb. Theory, Ser. A.

[27]  W. Harro,et al.  Upper bounds for Bermudan options on Markovian data using nonparametric regression and a reduced number of nested Monte Carlo steps , 2009 .

[28]  Arthur L. Besse,et al.  Einstein Manifolds and Topology , 1987 .

[29]  T. Merkle,et al.  Conformally closed Poincaré-Einstein metrics with intersecting scale singularities , 2008 .

[30]  Ingo Steinwart,et al.  Optimal regression rates for SVMs using Gaussian kernels , 2013 .

[31]  Ulrich Brehm,et al.  Lattice triangulations of E 3 and of the 3-torus , 2009 .

[32]  P. Sankaran.  Nonexistence of almost complex structures on Grassmann manifolds , 1991 .

[33]  Rupert L. Frank,et al.  Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain , 2011, 1105.5182.

[34]  M. Hamilton.  On certain exotic 4-manifolds of Akhmedov and Park , 2011, 1102.1376.

[35]  A. Borel,et al.  CHARACTERISTIC CLASSES AND HOMOGENEOUS SPACES, I.* , 1958 .

[36]  Hsien-chung Wang.  Closed Manifolds with Homogeneous Complex Structure , 1954 .

[37]  M Simon,et al.  k-NN Kernel Estimate for Nonparametric Functional Regression in Time Series Analysis , 2011 .

[38]  Leander Geisinger,et al.  Sharp spectral estimates in domains of infinite volume , 2010, 1012.5972.

[39]  Jens Wirth,et al.  Embedded Eigenvalues for an Elastic Strip with Cracks , 2012 .