In this paper, we obtain a localization formula in differential K-theory for $S^1$-action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $η$-invariants, we establish a version of localization formula for equivariant $η$-invariants. An important step of our approach is to construct a pre-$λ$-ring structure in differential K-theory which is interesting in its own right.
In this note, we give a short proof of the localization formula for the loop space Chern character of a compact Riemannian spin manifold M, using the rescaled spinor bundle on the tangent groupoid associated to M.
A Remark on the Localization formulas about two Killing vector fields
本文讨论零点集${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$上两个Killing向量场的等变上同调局部化公式。作为应用，我们利用它得到了有关示性数的公式，并得到了辛流形上的Duistermaat-Heckman型公式。
The Baker-Campbell-Hausdorff formula and the Zassenhaus formula in synthetic differential geometry
在Anders Kock [Taylor series calculus for ring objects of line type，Journal of Pure and Applied Algebra，12（1978），271-293]的启发下，我们将建立李群理论中的Baker-Campbell-Hausdorff公式和Zassenhaus公式。
We prove the Reilly formula for a class of elliptic divergence differential operator $L_Au=div(A\nabla u)$, where $A$ is a (1,1)-Codazzi tensor field. Then we get some estimates for the first positive eigenvalue of the operator.
