Jesús A. Álvarez López - Böcker

This book presents a new Lefschetz trace formula for a foliated flow on a compact foliated manifold with a foliation of codimension one. The leaves preserved by the flow and its closed orbits are assumed to be transversely simple. The formula equates two distributions on the real line: one is a renormalized trace of the flow’s action on two reduced leafwise cohomologies, defined via conormal and dual-conormal currents; the other consists of contributions from the preserved leaves, closed orbits, and a b-trace version of Connes’ Euler characteristic, defined using a transverse invariant measure on the complement of the preserved leaves. The usual Euler characteristic is undefined here due to non-compactness.  The proofs and definitions combine tools from analysis and foliation theory, like small b-calculus, Witten’s deformation of differential complexes, heat invariants, and zeta functions of operators, alongside a description of foliations with this type of flow. The exposition is largely self-contained, with prerequisites and references given for further study. The trace formula solves a conjecture of C. Deninger, motivated by his program which links arithmetic zeta functions to foliated geometry. While further generalization is needed for arithmetic applications, the original and technically deep ideas presented here are valuable in themselves and offer a foundation for future work.  The book will be of interest to researchers and graduate students in foliation theory, global analysis on manifolds, and arithmetic geometry.

This book provides a detailed introduction to the coarse quasi-isometry of leaves of a foliated space and describes the cases where the generic leaves have the same quasi-isometric invariants.

The subject of this volume, recent developments in foliation theory and important related analytic and geometric techniques, is an active field in the application of both global analysis and geometric topological theory of manifolds to the study of foliations. This volume includes research papers by leading specialists, giving an overview of this subject.

This volume contains research and expository papers on recent advances in foliations and Riemannian geometry. Some of the topics covered in this volume include: topology, geometry, dynamics and analysis of foliations, curvature, submanifold theory, Lie groups and harmonic maps.Among the contributions, readers may find an extensive survey on characteristic classes of Riemannian foliations offering also new results, an article showing the uniform simplicity of certain diffeomorphism groups, an exposition of convergences of contact structures to foliations from the point of view of Thurston's and Thurston-Bennequin's inequalities, a discussion about Fatou-Julia decompositions for foliations and a description of singular Riemannian foliations on spaces without conjugate points.Papers on submanifold theory focus on the existence of graphs with prescribed mean curvature and mean curvature flow for spacelike graphs, isometric and conformal deformations and detailed surveys on totally geodesic submanifolds in symmetric spaces, cohomogeneity one actions on hyperbolic spaces and rigidity of geodesic spheres in space forms. Geometric realizability of curvature tensors and curvature operators are also treated in this volume with special attention to the affine and the pseudo-Riemannian settings. Also, some contributions on biharmonic maps and submanifolds enrich the scope of this volume in providing an overview of different topics of current interest in differential geometry.