Cheeger-Chern-Simons classes of transversally symmetric foliations: Dependence relations and eta-invariants. (English) Zbl 0793.57015
The classical “Proportionality Principle” of F. Hirzebruch states that for two compact locally symmetric Hermitian spaces of the same non- compact type, the corresponding sets of Chern numbers are proportional with proportionality factor given by the ratio of the two volumes. For odd-dimensional manifolds all rational primary characteristic numbers are zero for trivial reasons, but instead there are well-defined secondary characteristic numbers of Cheeger-Chern-Simons type, the most important of which is the eta-invariant of Atiyah-Patodi-Singer. Analogously to the Hirzebruch Proportionality Principle we associate to the class of locally symmetric spaces with given universal covering a “degree of dependence” \(d\), and show that the set of secondary characteristic numbers for this class of manifolds is a rational vector space of at most dimension \(d\).
Reviewer: J.L.Dupont, F.W.Kamber (Aarhus)
MSC:
| 57R30 | Foliations in differential topology; geometric theory |
| 53C12 | Foliations (differential geometric aspects) |
| 57R20 | Characteristic classes and numbers in differential topology |
| 53C35 | Differential geometry of symmetric spaces |
Keywords:
proportionality principle; degree of dependence; Cheeger-Chern-Simons class; transversally symmetric foliation; compact locally symmetric Hermitian spaces; secondary characteristic numbers; eta-invariantReferences:
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