Volterra calculus and the variation formulas for the equivariant Ray–Singer metric

In this paper, using the Greiner's approach to heat kernel asymptotics, we give new proofs of the equivariant Gauss–Bonnet–Chern formula and the variation formulas for the equivariant Ray–Singer metric, which are originally due to Bismut and Zhang.

The Noncommutative Infinitesimal Equivariant Index Formula

In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regularity and give an explicit formula for them. We also establish an infinitesimal equivariant family index formula and introduce the infinitesimal equivariant eta forms as well as compare them with the equivariant eta forms.