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}) | + | 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 $n$-dimensional spin manifold (compare [[Spin bordism|Spin bordism Invariants]]) is given as follows: Let $Spin(M)$ the principal $Spin(n)$-bundle of $M$, and let $S$ be obtained by adjoining the real Clifford algebra $Cl(n)$ to $Spin(M)$ using the left multiplication of elements in $Spin(n)$ on $Cl(n)$. The Dirac operator $D$ then acts on the space of sections $\Gamma(S)$. The kernel of $D$ is called the space of (real) harmonic spinors. In case $\dim(M)=1 \mod 8$ the space of harmonic spinors canonically has the structure of a complex vector space, while |
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− | The $\alpha$-invariant for a closed $n$-dimensional spin manifold (compare [[Spin bordism|Spin bordism Invariants]]) is given as follows: Let $Spin(M)$ the principal $Spin(n)$-bundle of $M$, and let $S$ be obtained by adjoining the real Clifford algebra $Cl(n)$ to $Spin(M)$ using the left multiplication of elements in $Spin(n)$ on $Cl(n)$. The Dirac operator $D$ then acts on the space of sections $\Gamma(S)$. The kernel of $D$ is called the space of (real) harmonic spinors. 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 space of harmonic spinors yields an element in $KO_*$, the $\alpha$-invariant. In particular, if the $\alpha$-invariant is non-trivial, the operator $D$ must have a non-trivial kernel. | in case $\dim(M)=2 \mod 8$ the space of harmonic spinors canonically carries the structure of a quarternionic vector space. The space of harmonic spinors yields an element in $KO_*$, the $\alpha$-invariant. In particular, if the $\alpha$-invariant is non-trivial, the operator $D$ must have a non-trivial kernel. | ||
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{{endthm}} | {{endthm}} | ||
− | The fact that $\alpha(\Sigma)$ is an obstruction follows form the [wiki/Weitzenböck_identity|Bochner-Weitzenböck formula], which yields the formula $D^2 = \ | + | The fact that $\alpha(\Sigma)$ is an obstruction follows form the [wiki/Weitzenböck_identity|Bochner-Weitzenböck formula], which yields the formula $D^2 = \nabla^*\nabla + \frac{1}{4}scal(M)$ for the Dirac operator $D$. Here $\nabla^*\nabla$ denotes the connection Laplacian which is a non-negative operator. Hence, if the scalar curature function $scal(M)$ is strictly positive the operator $D$ cannot have a non-trivial kernel. |
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1 Introduction
This page is in the construction process!!!
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 closed -dimensional spin manifold (compare Spin bordism Invariants) is given as follows: Let the principal -bundle of , and let be obtained by adjoining the real Clifford algebra to using the left multiplication of elements in on . The Dirac operator then acts on the space of sections . The kernel of is called the space of (real) harmonic spinors. 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 space of harmonic spinors yields an element in , the -invariant. In particular, if the -invariant is non-trivial, the operator must have a non-trivial kernel.
Theorem 7.1. Let be an -dimensional homotopy sphere with then admits a metric of positive scalar curvature if and only if is trivial.
The -invariant of a homotopy sphere can be computed explicitely by the following means; therefore note that is isomorphic to for or .
Proposition 7.2. The -invariant of a homotopy sphere is given by
The fact that is an obstruction follows form the [wiki/Weitzenböck_identity|Bochner-Weitzenböck formula], which yields the formula for the Dirac operator . Here denotes the connection Laplacian which is a non-negative operator. Hence, if the scalar curature function is strictly positive the operator cannot have a non-trivial kernel.
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