Curvature properties of exotic spheres
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More information on the $\alpah$-invariant can be found at [[Spin | More information on the $\alpah$-invariant can be found at [[Spin | ||
− | 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. | + | 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. | ||
The $\alpha$-invariant of a homotopy sphere $\Sigma$ is given by | The $\alpha$-invariant of a homotopy sphere $\Sigma$ is given by |
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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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