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. 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 spin manifold is given as follows: ... |
− | More information on the $\ | + | More information on the $\alpha$-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 $\dim(M)=1 \mod 8$ 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 $\dim(M)=2 \mod 8$ the space of harmonic spinors canonically carries the structure of a quarternionic vector space. | ||
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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 (cf. [Hitchin1974], [Lichnerowicz1963]). The -invariant for a spin manifold 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