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2 results

  • On a generalized conjecture of Hopf with symmetry

  • Part of
  • Manuel Amann, Lee Kennard
  • Journal:
    Compositio Mathematica / Volume 153 / Issue 2 / February 2017
    Published online by Cambridge University Press:
    01 February 2017, pp. 313-322
    Print publication:
    February 2017
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  • A famous conjecture of Hopf states that $\mathbb{S}^{2}\times \mathbb{S}^{2}$ does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product $N\times N$ can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.

  • The first Pontryagin class

  • Paul Bressler
  • Journal:
    Compositio Mathematica / Volume 143 / Issue 5 / September 2007
    Published online by Cambridge University Press:
    20 September 2007, pp. 1127-1163
    Print publication:
    September 2007
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  • We give a natural obstruction theoretic interpretation to the first Pontryagin class in terms of Courant algebroids. As an application we calculate the class of the stack of algebras of chiral differential operators. In particular, we establish the existence and uniqueness of the chiral de Rham complex.