Curvature properties of exotic spheres
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== Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == | == Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == | ||
− | 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 (cf. \cite{Hitchin1974}, \cite{Lichnerowicz1963}). The $\alpha$-invariant for a spin manifold is given as follows: | + | 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 (cf. \cite{Hitchin1974}, \cite{Lichnerowicz1963}). The $\alpha$-invariant for a closed spin manifold (compare [[Spin bordism|Spin bordism Invariants]]) is given as follows: |
More information on the $\alpha$-invariant can be found at [[Spin | More information on the $\alpha$-invariant can be found at [[Spin | ||
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The $\alpha$-invariant of a homotopy sphere $\Sigma$ is given by | The $\alpha$-invariant of a homotopy sphere $\Sigma$ is given by | ||
− | $$ \alpha(Sigma) = \left\{\begin{array}{lcl} | + | $$ \alpha(\Sigma) = \left\{\begin{array}{lcl} |
− | \dim_{\ | + | \dim_{\Cc}Ker(D) \mod 2 && \text{if $\dim(\Sigma)= 1 \mod 8\\ |
− | \dim_{\ | + | \dim_{\Hh}Ker(D) \mod 2 && \text{if $\dim(\Sigma)= 2 \mod 8\\ |
0 && in all other cases\right.$$ | 0 && in all other cases\right.$$ | ||
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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 - of a closed spin manifold provides an obstruction to the existence of a metric of positive scalar curvature on it (cf. [Hitchin1974], [Lichnerowicz1963]). The -invariant for a closed spin manifold (compare Spin bordism Invariants) is given as follows: More information on the -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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2 References
- [Hitchin1974] N. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR0358873 (50 #11332) Zbl 0284.58016
- [Lichnerowicz1963] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9. MR0156292 (27 #6218) Zbl 0714.53041