The first eigenvalue of the Dirac operator on compact spin symmetric spaces. (English) Zbl 1079.53068
Let \(M=G/K\) be a compact, simply connected, irreducible Riemannian symmetric space endowed with the Killing metric and \(n = \dim M\). Assume that \(G\) and \(K\) have the same rank and that \(M\) has a spin structure. The author proves that the square of the first eigenvalue of the Dirac operator is equal to \(2 \min_{1\leq k \leq p}\| \beta_k\| ^2 + n/8\), where \(\beta_1,\ldots\beta_p\) are the \(K\)-dominant weights occurring in the decomposition into irreducible components of the spin representation under the action of \(K\), and the norm is the one induced from the Killing form.
Reviewer: Jürgen Berndt (Cork)
MSC:
| 53C27 | Spin and Spin\({}^c\) geometry |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 53C35 | Differential geometry of symmetric spaces |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
References:
| [1] | Bär, C.: Das Spektrum von Dirac-Operatoren. Dissertation, Universität Bonn, 1991, Bonner Mathematische Schriften 217. |
| [2] | Besse, A.: Einstein Manifolds. Berlin: Springer-Verlag, 1987 · Zbl 0613.53001 |
| [3] | Bourguignon, J.P., Hijazi, O., Milhorat, J.-L., Moroianu, A.: A Spinorial Approach to Riemannian and Conformal Geometry. Monograph (in preparation) · Zbl 1348.53001 |
| [4] | Cahen, M., Franc, A., Gutt, S.: Spectrum of the Dirac Operator on Complex Projective Space P2q-1(). Lett. Math. Phys. 18, 165–176 (1989) · Zbl 0706.58010 · doi:10.1007/BF00401871 |
| [5] | Cahen, M., Gutt, S.: Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces. Simon Stevin 62, 209–242 (1988) · Zbl 0677.53057 |
| [6] | Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied mathematics, Vol. 80. San Diego: Academic Press, 1978 · Zbl 0451.53038 |
| [7] | Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Berlin-Heidelberg New York: Springer-Verlag, 1972 · Zbl 0254.17004 |
| [8] | Parthasarathy, R.: Dirac operator and the discrete series. Ann. Math. 96, 1–30 (1971) · Zbl 0249.22003 · doi:10.2307/1970892 |
| [9] | Sulanke, S.: Die Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre. Doktorarbeit, Humboldt-Universität, Berlin, 1979 |
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