Real Hyperbolic Manifolds Research Articles

Exotic Smoothings of Hyperbolic Manifolds which do not Support Pinched Negative Curvature

Constructed in this note are examples of topological manifolds M supporting at least two distinct smooth structures ${\mathcal {M}_1}$ and ${\mathcal {M}_2}$, where ${\mathcal {M}_1}$ is a complete, finite volume, real hyperbolic manifold, while ${\mathcal {M}_2}$ cannot support a complete, finite volume, pinched negatively curved Riemannian metric.

Exotic smoothings of hyperbolic manifolds which do not support pinched negative curvature

Constructed in this note are examples of topological manifolds M supporting at least two distinct smooth structures M 1 {\mathcal {M}_1} and M 2 {\mathcal {M}_2} , where M 1 {\mathcal {M}_1} is a complete, finite volume, real hyperbolic manifold, while M 2 {\mathcal {M}_2} cannot support a complete, finite volume, pinched negatively curved Riemannian metric.

Negatively Curved Manifolds with Exotic Smooth Structures

Let $M$ denote a compact real hyperbolic manifold with dimension $m \geq 5$ and sectional curvature $K = - 1$, and let $\Sigma$ be an exotic sphere of dimension $m$. Given any small number $\delta > 0$, we show that there is a finite covering space $\widehat {M}$ of $M$ satisfying the following properties: the connected sum $\widehat {M}\# \Sigma$ is not diffeomorphic to $\widehat {M}$, but it is homeomorphic to $\widehat {M}$; $\widehat {M}\# \Sigma$ supports a Riemannian metric having all of its sectional curvature values in the interval $[ - 1 - \delta , - 1 + \delta ]$. 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 Lawson-Yau conjecture. Note that Mostow’s Rigidity Theorem [17] implies that $\widehat {M}\# \Sigma$ 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 $\widehat {M}\# \Sigma$ 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].

Negatively curved manifolds with exotic smooth structures

Let M M denote a compact real hyperbolic manifold with dimension m ≥ 5 m \geq 5 and sectional curvature K = − 1 K = - 1 , and let Σ \Sigma be an exotic sphere of dimension m m . Given any small number δ > 0 \delta > 0 , we show that there is a finite covering space M ^ \widehat {M} of M M satisfying the following properties: the connected sum M ^ # Σ \widehat {M}\# \Sigma is not diffeomorphic to M ^ \widehat {M} , but it is homeomorphic to M ^ \widehat {M} ; M ^ # Σ \widehat {M}\# \Sigma supports a Riemannian metric having all of its sectional curvature values in the interval [ − 1 − δ , − 1 + δ ] [ - 1 - \delta , - 1 + \delta ] . 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 Lawson-Yau conjecture. Note that Mostow’s Rigidity Theorem [17] implies that M ^ # Σ \widehat {M}\# \Sigma 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 M ^ # Σ \widehat {M}\# \Sigma 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].