An anomaly formula for Ray–Singer metrics on manifolds with boundary

Abstract

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...

The Mathematical Work of V. K. Patodi

“…The RaySinger torsion for manifolds with boundary has been studied extensively. Here we only mention two papers of Brüning-Ma [57,58], which deal with the most general case of arbitrary flat vector bundles as well as arbitrary metrics near the boundary. In particular, a local Gauss-Bonnet-Chern theorem, generalizing the work of Patodi [2] to the case of manifolds with boundary, is established in [57], where no condition on metrics near the boundary is assumed.…”

Torção de Reidemeister das formas espaciais esféricas

“…No entanto, surgiriam algumas dificuldades técnicas que estão além do objetivo deste trabalho. Como observado na Introdução, existem duas fórmulas distintas para tal razão, uma no Teorema 1 de [10] e a outra no Teorema 1 de [5], que será a utilizada aqui.…”

“…Primeiro, obtemos do Lema 4.11 uma fórmula para a razão em termos de alguns invariantes geométricos. Isso segue do Teorema 1 de [5] e nos dá, no caso de dimensão ímpar, a razão em termos da característica de Euler do bordo.…”

“…O caso de dimensão par é bem mais complicado e necessita de algumas notações e ferramentas apresentadas em [3,5]. Segundo, o Lema 4.12 fornece o cálculo da integral que aparece na segunda equação do Lema 4.11, concluindo assim o caso de dimensão par.…”

“…onde as formas B(∇ T X j ) estão definidas na equação (1.17) de [5]. Desde que g 0 é um produto próximo ao bordo, pelo resultado de [19] temos…”