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Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the sixteenth century. Its phylogeny and history are presented in the Prologue of this volume.
The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives.
Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, Weitzenböck formulas, and Finslerian spinors and Dirac operators. The tools developed in this book will find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua.
Audience: This text will be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua.
List of contents
Section I. Finsler Laplacians in Application.- to Diffusions on Finsler Manifolds.- Density Dependent Host/Parasite Systems of Rothschild Type and Finslerian Diffusion.- Stochastic Finsler Geometry in the Theory of Evolution by Symbiosis.- Section II. Stochastic Analysis and Brownian Motion.- Diffusions on Finsler Manifolds.- Stochastic Calculus on Finsler Manifolds and an Application in Biology.- Diffusion on the Tangent and Indicatrix Bundles of a Finsler Manifold.- Section III. Stochastic Lagrange Geometry.- Diffusion on the Total Space of a Vector Bundle.- Diffusions and Laplacians on Lagrange Manifolds.- ?-Lagrange Laplacians.- Section IV. Mean-Value Properties of Harmonic Functions.- Diffusion, Laplacian and Hodge Decomposition on Finsler Spaces.- A Mean-Value Laplacian for Finsler Spaces.- Section V. Analytical Constructions.- The Non-Linear Laplacian for Finsler Manifolds.- A Bochner Vanishing Theorem for Elliptic Complices.- A Lichnerowicz Vanishing Theorem for Finsler Spaces.- A Geometric Inequality and a Weitzenböck Formula for Finsler Surfaces.- Spinors on Finsler Spaces.
