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On the Topology of Scalar‐Flat Manifolds
Author(s) -
Dessai Anand
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/33.2.203
Subject(s) - mathematics , scalar curvature , scalar (mathematics) , pure mathematics , manifold (fluid mechanics) , prescribed scalar curvature problem , curvature , cohomology , metric (unit) , simply connected space , cohomology ring , dimension (graph theory) , topology (electrical circuits) , mathematical analysis , sectional curvature , geometry , combinatorics , equivariant cohomology , mechanical engineering , operations management , engineering , economics
Let M be a simply connected closed manifold of dimension greater than 4 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar‐flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M . As a consequence, any simply connected scalar‐flat manifold of dimension greater than 4 with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar‐flat metrics, almost complex structures and the free loop space.