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It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent.
A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a leading pioneer in the field, it focuses on the functorial and special cases methods of computi
This book is a collection of lecture notes and survey papers based on the minicourses given by leading experts at the 2016 CRM Summer School on Spectral Theory and Applications, held from July 4–14, 2016, at Université Laval, Québec City, Québec, Canada. The papers contained in the volume cover a broad variety of topics in spectral theory, starting from the fundamentals and highlighting its connections to PDEs, geometry, physics, and numerical analysis.
Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more.
The Marcel Grossmann meetings were conceived to promote theoretical understanding in the fields of physics, mathematics, astronomy and astrophysics and to direct future technological, observational, and experimental efforts. They review recent developments in gravitation and general relativity, with major emphasis on mathematical foundations and physical predictions. Their main objective is to bring together scientists from diverse backgrounds and their range of topics is broad, from more abstract classical theory and quantum gravity and strings to more concrete relativistic astrophysics observations and modeling.This Tenth Marcel Grossmann Meeting was organized by an international committee composed of D Blair, Y Choquet-Bruhat, D Christodoulou, T Damour, J Ehlers, F Everitt, Fang Li Zhi, S Hawking, Y Ne'eman, R Ruffini (chair), H Sato, R Sunyaev, and S Weinberg and backed by an international coordinating committee of about 135 members from scientific institutions representing 54 countries. The scientific program included 29 morning plenary talks during 6 days, and 57 parallel sessions over five afternoons, during which roughly 500 papers were presented.These three volumes of the proceedings of MG10 give a broad view of all aspects of gravitation, from mathematical issues to recent observations and experiments.
The volume will consist of about 40 articles written by some very influential mathematicians of our time and will expose the latest achievements in the broad area of nonlinear analysis and its various interdisciplinary applications.
Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications is a rich and self-contained exposition of recent developments in Riemannian submersions and maps relevant to complex geometry, focusing particularly on novel submersions, Hermitian manifolds, and K\{a}hlerian manifolds. Riemannian submersions have long been an effective tool to obtain new manifolds and compare certain manifolds within differential geometry. For complex cases, only holomorphic submersions function appropriately, as discussed at length in Falcitelli, Ianus and Pastore's classic 2004 book. In this new book, Bayram Sahin extends the scope of complex cases with wholly new submersion types, including Anti-invariant submersions, Semi-invariant submersions, slant submersions, and Pointwise slant submersions, also extending their use in Riemannian maps. The work obtains new properties of the domain and target manifolds and investigates the harmonicity and geodesicity conditions for such maps. It also relates these maps with discoveries in pseudo-harmonic maps. Results included in this volume should stimulate future research on Riemannian submersions and Riemannian maps. - Systematically reviews and references modern literature in Riemannian maps - Provides rigorous mathematical theory with applications - Presented in an accessible reading style with motivating examples that help the reader rapidly progress
This book gives a presentation of topics in Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject. The authors have aimed at presenting technical material in a clear and detailed manner. In this volume, geometric aspects of the theory have been emphasized. The book presents the theory of Ricci solitons, Kahler-Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local collapsing, Perelman's reduced distance function and applications to ancient solutions, and a primer of 3-manifold topology. Various technical aspects of Ricci flow have been explained in a clear and detailed manner. The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to Perelman's work and explains technical aspects of Ricci flow useful for singularity analysis. Throughout, there are appropriate references so that the reader may further pursue the statements and proofs of the various results.
A brief portrait of the life and work of Professor Enrique Vidal Abascal / L.A. Cordero -- pt. A. Foliation theory. Characteristic classes for Riemannian foliations / S. Hurder. Non unique-ergodicity of harmonic measures: Smoothing Samuel Petite's examples / B, Deroin. On the uniform simplicity of diffeomorphism groups / T. Tsuboi. On Bennequin's isotopy lemma and Thurston's inequality / Y. Mitsumatsu. On the Julia sets of complex codimension-one transversally holomorphic foliations / T. Asuke. Singular Riemannian foliations on spaces without conjugate points / A. Lytchak. Variational formulae for the total mean curvatures of a codimension-one distribution / V. Rovenski and P. Walczak. On a Weitzenböck-like formula for Riemannian foliations / V. Slesar. Duality and minimality for Riemannian foliations on open manifolds / X.M. Masa. Open problems on foliations -- pt. B. Riemannian geometry. Graphs with prescribed mean curvature / M. Dajczer. Genuine isometric and conformal deformations of submanifolds / R. Tojeiro. Totally geodesic submanifolds in Riemannian symmetric spaces / S. Klein. The orbits of cohomogeneity one actions on complex hyperbolic spaces / J.C. Díaz-Ramos. Rigidity results for geodesic spheres in space forms / J. Roth. Mean curvature flow and Bernstein-Calabi results for spacelike graphs / G. Li and I.M.C. Salavessa. Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators / P. Gilkey, S. Nikc̮ević and D. Westerman. Conformally Osserman multiply warped product structures in the Riemannian setting / M. Brozos-Vázquez, M.E. Vázquez-Abal and R. Vázquez-Lorenzo. Riemannian [symbol]-symmetric spaces / M. Goze and E. Remm. Methods for solving the Jacobi equation. Constant osculating rank vs. constant Jacobi osculating rank / T. Arias-Marco. On the reparametrization of affine homogeneous geodesics / Z. Dus̮ek. Conjugate connections and differential equations on infinite dimensional manifolds / M. Aghasi [und weitere]. Totally biharmonic submanifolds / D. Impera and S. Montaldo. The biharmonicity of unit vector fields on the Poincaré half-space H[symbol] / M.K. Markellos. Perspectives on biharmonic maps and submanifolds / A. Balmus. Contact pair structures and associated metrics / G. Bande and A. Hadjar. Paraquaternionic manifolds and mixed 3-structures / S. Ianus and G.E. Vi̮lcu. On topological obstruction of compact positively Ricci curved manifolds / W.-H. Chen. Gray curvature conditions and the Tanaka-Webster connection / R. Mocanu. Riemannian structures on higher order frame bundles from classical linear connections / J. Kurek and W.M. Mikulski. Distributions on the cotangent bundle from torsion-free connections / J. Kurek and W.M. Mikulski. On the geodesics of the rotational surfaces in the Bianchi-Cartan-Vranceanu spaces / P. Piu and M.M. Profir. Cotangent bundles with general natural Kähler structures of quasi-constant holomorphic sectional curvatures / S.L. Druta̮. Polynomial translation Weingarten surfaces in 3-dimensional Euclidean space / M.I. Munteanu and A.I. Nistor. G-structures defined on pseudo-Riemannian manifolds / I. Sánchez-Rodríguez -- List of participants