Curvature properties of exotic spheres
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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 results in [Lichnerowicz1963]) 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]). 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 determines an element in , the -invariant; and in particular, if the -invariant is non-trivial, the operator must have a non-trivial kernel.
The -invariant of a homotopy sphere can be computed explicitely by the following means; therefore note that is isomorphic to for or .
Proposition 7.1. The -invariant of a homotopy sphere is given by
Theorem 7.2. Let be an -dimensional homotopy sphere with then admits a metric of positive scalar curvature if and only if is trivial.
The fact that -invariant is an obstruction to the existence of postive scalar curvature follows from the 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, thus the -invariant must be trivial.
On the other hand, Stolz in [Stolz1992] proved that a simply connected closed spin manifold of dimension admits a metric of positive scalar curvature if its -invariant vanishes. The proof uses the surgery results for scalar curvature obtained (independently) in [Gromov&Lawson1980] and [Schoen&Yau1979], as well as a quite involved calculation within stable homotopy theory.
- [Gromov&Lawson1980] M. Gromov and H. B. Lawson, Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. (2) 111 (1980), no.2, 209–230. MR569070 (81g:53022) Zbl 0445.53025
- [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
- [Schoen&Yau1979] R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no.1-3, 159–183. MR535700 (80k:53064) Zbl 0423.53032
- [Stolz1992] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no.3, 511–540. MR1189863 (93i:57033) Zbl 0784.53029