Abstract. Using the heat kernel, we derive first a local Gauss-BonnetChern theorem for manifolds with a non-product metric near the boundary. Then we establish an anomaly formula for Ray-Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.
IntroductionLet X be a compact m-dimensional smooth manifold, not necessarily oriented, with boundary ∂X = Y and canonical embedding j : Y → X, and let F be a flat complex vector bundle over X, equipped with its flat connection ∇ F . We denote by H • (X, F ) = m p=0 H p (X, F ) the (absolute) de-Rham cohomology of X with coefficients in F . For a finite dimensional vector space E, set det E := Λ max E, and denote by (det E) −1 := det E * the dual line. Then it is customary to call the complex line, [H], under the assumption that h F is flat and that g T X has product structure near the boundary. Dai and Fang [DF] were the first to study this problem with h F flat but without assuming a product structure for g T X near Y ; they used methods completely different from ours. In [LüS], Lück and Schick computed some examples to show that the boundary term does not vanish in general. We learned about this problem from [DF] which was very helpful for us even though we investigate the problem from a rather different perspective. As an interesting by-product, our reasoning also leads to a local Gauss-Bonnet-Chern theorem [Ch1,2], see Theorem 3.2 below, for manifolds without a product structure near the boundary. As a notable feature of our proof, we derive the boundary contribution from the fundamental solution of the model problem.Let us now describe the geometric setting in greater detail. Let
) is closed in Ω(X, Y, o(T X)) and defines the relative Euler class of T X, and E(T X) := [E(T X, ∇ T X )] ∈ H m (X, Y, o(T X)) does not depend on the choice of g T X .Next let (g T X 0 , h F 0 ) and (g T X 1 , h F 1 ) be two couples of metrics on (T X, F ). We will use the subscripts 0, 1 to distinguish various objects associated with these metrics. In Theorem 1.9, we show that there exists a canonical classIt is obvious from (0.4) that E defines the secondary relative Euler class of T X in the spirit of Chern-Simons theory.be the closed 1-form (cf. Definition 4.4), which measures the variation of the metric det F,1 on det F with respect to the induced flat connection on det F ; θ(F, h F 1 ) vanishes if the metric det F,1 is flat. The main result of this paper generalizes [BiZ1, Thm. 0.1] to manifolds with boundary; it reads as follows: In fact, in [Ch, Cor. 3.29] Cheeger states (what is also implicit in the proof of [RS, Thm. 7.3]) that the left-hand side of (0.6) depends only on the germ of g T X i restricted to Y , a crucial fact for Cheeger's proof of the Ray-Singer conjecture [Ch, §4]. In subsection 4.5, we find that our results coincide with the results in [LüS, App. A] where Lüc...
