Negatively curved manifolds with exotic smooth structures
Authors:
F. T. Farrell and L. E. Jones
Journal:
J. Amer. Math. Soc. 2 (1989), 899908
MSC:
Primary 53C20; Secondary 57R10, 57R55, 57R67
MathSciNet review:
1002632
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Abstract: Let denote a compact real hyperbolic manifold with dimension and sectional curvature , and let be an exotic sphere of dimension . Given any small number , we show that there is a finite covering space of satisfying the following properties: the connected sum is not diffeomorphic to , but it is homeomorphic to ; supports a Riemannian metric having all of its sectional curvature values in the interval . Thus, there are compact Riemannian manifolds of strictly negative sectional curvature which are not diffeomorphic but whose fundamental groups are isomorphic. This answers Problem 12 of the list compiled by Yau [22]; i.e., it gives counterexamples to the LawsonYau conjecture. Note that Mostow's Rigidity Theorem [17] implies that does not support a Riemannian metric whose sectional curvature is identically 1 . (In fact, it is not diffeomorphic to any locally symmetric space.) Thus, the manifold supports a Riemannian metric with sectional curvature arbitrarily close to 1 , but it does not support a Riemannian metric whose sectional curvature is identically 1 . More complicated examples of manifolds satisfying the properties of the previous sentence were first constructed by Gromov and Thurston [11].
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 R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 149. MR 0251664 (40:4891)
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 [5]
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 , Compact negatively curved manifolds (of ) are topologically rigid, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 34613463. MR 997635 (90h:57023b)
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 , A topological analogue of Mostow's Rigidity Theorem, J. Amer. Math. Soc. 2 (1989), 257370. MR 973309 (90h:57023a)
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 , Examples of expanding endomorphisms on exotic tori, Invent. Math. 45 (1978), 175179. MR 0474416 (57:14056)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S08940347198910026322
PII:
S 08940347(1989)10026322
Article copyright:
© Copyright 1989
American Mathematical Society
