zbMATH — the first resource for mathematics

Local invariants of spectral asymmetry. (English) Zbl 0538.58038
The eta invariant of Atiyah-Patodi-Singer is a spectral invariant measuring the spectral asymmetry of a Riemannian manifold. A priori the value of eta at \(s=0\) is not regular and one must prove that the local pole integrates to zero. The residue is an example of a spectral invariant given by a local formula which vanishes on definite operators and which is additive with respect to direct sums. In the present paper, the author considers slightly more general such invariants and manages to prove that they all vanish in even dimensions and on certain odd dimensional manifolds. He also corrects some errors in an earlier paper discussing the regularity of the global eta function at \(s=0\).
Reviewer: P.Gilkey

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
Full Text: DOI EuDML
[1] Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations. Invent. Math.50, 219-248 (1979) · Zbl 0393.35058 · doi:10.1007/BF01410079
[2] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Camb. Phil. Soc.79, 71-99 (1976) · Zbl 0325.58015 · doi:10.1017/S0305004100052105
[3] Gilkey, P.B.: Smooth invariants of a riemannian manifold. Adv. Math.28, 1-10 (1978) · Zbl 0378.53006 · doi:10.1016/0001-8708(78)90043-9
[4] Gilkey, P.B.: The residue of the local eta-function at the origin. Math. Ann.240, 183-189 (1979) · Zbl 0405.58045 · doi:10.1007/BF01364633
[5] Gilkey, P.B.: The residue of the global ?-function at the origin. Adv. Math.40, 290-307 (1981) · Zbl 0469.58015 · doi:10.1016/S0001-8708(81)80007-2
[6] Karoubi, M.:K-theory. Berlin-Heidelberg-New York: Springer 1978 · Zbl 0382.55002
[7] Lebiediev, D.R., Manin, Yu.I.: The Gelfand-Dikii Hamiltonian operator and the co-adjoint representation of the Volterra group. Funct. Anal. Appl.13, 40-46 (1976) (Russian); 268-273 (English)
[8] Seeley, R.T.: Complex powers of an elliptic operator. Proc. Sympos. Pure Math.10, Amer. Math. Soc. 288-307, 1967 · Zbl 0159.15504
[9] Spanier, E.H.: Algebraic topology. New York: McGraw-Hill 1966 · Zbl 0145.43303
[10] Stong, R.E.: Notes on cobordism theory. Princeton, Univ. Press 1968 · Zbl 0181.26604
[11] Wodzicki, M.: Spectral asymmetry and zeta functions. Invent. Math.66, 115-135 (1982) · Zbl 0489.58030 · doi:10.1007/BF01404760
[12] Manin, Yu.I.: Algebraic aspects of non-linear differential equations. Itogi Nauki i Tekhniki, ser. Sovremennyje Problemy Matematiki11, 5-152 (1978) (Russian); J. Sov. Math.11, 1-122 (1979) (English)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.