ETA INVARIANTS FOR MANIFOLD WITH BOUNDARY

Abstract:

For a compact manifold with boundary, M, there are well known local boundary conditions that make the de Rham operator A = d+δ elliptic, namely the absolute and relative boundary conditions. We study the eta invariants of such elliptic boundary value problems under the metric deformation

where x ɛ C^∞(M) is, near the boundary, the geodesic distance to the boundary, and g is a Riemannian metric on M which is of product type near the boundary. Under certain acyclicity condition we show that when M is odd dimensional where the subscript a (r) indicates the absolute (relative) boundary condition, and η_b(A₀) is the b-eta invariant of the limiting operator A₀. If M is even dimensional then Most of the analysis extends to analytic torsion, yielding when dim M is odd, and when dim M is even. Here the sign ± depends on the choice of the boundary condition and r₁ , r₂ vanishes at ɛ = 0.