Riemannian foliations of bounded geometry

Abstract

Abstract. Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will given in a forthcoming paper.

Modified Novikov Operators and the Kastler-Kalau-Walze-Type Theorem for Manifolds with Boundary

“…Let V ∈ X(M ) be determined by g(V, ·) = θ. Then we have the Novikov operators defined by θ, depending on z ∈ C in [16], where Rez is the real part of z, Imz is the imaginary part of z,c(θ) = (θ) * ∧ +(θ∧) * , c(θ) = (θ) * ∧ −(θ∧) * .…”

“…In others words, Wang provides a kind of method to study the Kastler-Kalau-Walze type theorem for manifolds with boundary. In [16], López and his collaborators introduced an elliptic differential operator which is called Novikov operator. The motivation of this paper is to prove the Kastler-Kalau-Walze type theorem for Novikov operators on manifolds with boundary.…”

The Adiabatic Limit of the Connection Laplacian

“…Remark 2.6. A related concept is that of a foliation of bounded geometry, introduced by Sanguiao [San08] and given a coordinate-free form by Álvarez Lopez, Kordyukov and Leichtnam [ALKL14]. When ∂M = ∅ = ∂F , Definition 2.5 implies that (M, g) with its foliation into fibres is a foliation of bounded geometry in the sense of [San08,ALKL14].…”

“…A.7]. However, bounded geometry in the sense of [San08,ALKL14] does not imply uniform local trivialisability, because the choice of a fixed reference metric g F on F gives global (upper) bounds on quantities such as the volume of M x , whereas the purely local definition of [San08,ALKL14] cannot achieve this. With all the necessary definitions at hand, we can now summarise our conditions on the geometry: Condition 2.…”

“…A related concept is that of a foliation of bounded geometry, introduced by Sanguiao[San08] and given a coordinate-free form by Álvarez Lopez, Kordyukov and Leichtnam[ALKL14]. When ∂M = ∅ = ∂F , Definition 2.5 implies that (M, g) with its foliation into fibres is a foliation of bounded geometry in the sense of[San08,ALKL14]. The trivialisations Φ of M provide suitable coordinate systems of product form, whose size can be estimated by the uniform lower bound on the injectivity radius of M x (with the induced metric) given by [Lam14, Lem.…”