Curvature properties of exotic spheres
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manifold which is homotopy equivalent to the standard sphere $S^n$. A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. | manifold which is homotopy equivalent to the standard sphere $S^n$. A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. | ||
General information about homotopy spheres (and exotic spheres) is given on [[Exotic spheres]]. One prominent question concerning the geometry of exotic spheres is: | General information about homotopy spheres (and exotic spheres) is given on [[Exotic spheres]]. One prominent question concerning the geometry of exotic spheres is: | ||
− | Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? | + | Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? One typically considers the following three types. |
== Homotopy spheres with positive [[Wikipedia:Sectional_curvature|sectional curvature]] == | == Homotopy spheres with positive [[Wikipedia:Sectional_curvature|sectional curvature]] == | ||
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== Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == | == Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == | ||
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+ | Hitchin (based on work of Lichnerowicz) proved that the so-called $\alpha$-invariant of a closed spin manifold provides an obstruction to the existence of a metric of positive scalar curvature on it. The $\alpha$-invariant for a spin manifold is given as follows: ... | ||
+ | More information on the $\alpah$-invariant can be found at [[Spin | ||
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+ | In case $\dim(M)=1 \mod 8$ the space of harmonic spinors canonically has the structure of a complex vector space, while in case $\dim(M)=2 \mod 8$ the space of harmonic spinors canonically carries the structure of a quarternionic vector space. | ||
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+ | The $\alpha$-invariant of a homotopy sphere $\Sigma$ is given by | ||
+ | $$ \alpha(Sigma) = \left\{\begin{array}{lcl} | ||
+ | \dim_{\CC}Ker(D) \mod 2 && \text{if $\dim(\Sigma)= 1 \mod 8\\ | ||
+ | \dim_{\HH}Ker(D) \mod 2 && \text{if $\dim(\Sigma)= 2 \mod 8\\ | ||
+ | 0 && in all other cases\right.$$ | ||
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1 Introduction
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A homotopy sphere of dimension is an oriented closed smooth manifold which is homotopy equivalent to the standard sphere . A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. General information about homotopy spheres (and exotic spheres) is given on Exotic spheres. One prominent question concerning the geometry of exotic spheres is: Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? One typically considers the following three types.
1 Homotopy spheres with positive sectional curvature
2 Homotopy spheres with positive Ricci curvature
3 Homotopy spheres with positive scalar curvature
Hitchin (based on work of Lichnerowicz) proved that the so-called -invariant of a closed spin manifold provides an obstruction to the existence of a metric of positive scalar curvature on it. The -invariant for a spin manifold is given as follows: ...
More information on theTex syntax error-invariant can be found at [[Spin
In case the space of harmonic spinors canonically has the structure of a complex vector space, while in case the space of harmonic spinors canonically carries the structure of a quarternionic vector space.
The -invariant of a homotopy sphere is given by
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