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Comparison Finsler Geometry (Paperback, 1st ed. 2021)
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Comparison Finsler Geometry (Paperback, 1st ed. 2021)
Series: Springer Monographs in Mathematics
Expected to ship within 12 - 17 working days
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This monograph presents recent developments in comparison geometry
and geometric analysis on Finsler manifolds. Generalizing the
weighted Ricci curvature into the Finsler setting, the author
systematically derives the fundamental geometric and analytic
inequalities in the Finsler context. Relying only upon knowledge of
differentiable manifolds, this treatment offers an accessible entry
point to Finsler geometry for readers new to the area. Divided into
three parts, the book begins by establishing the fundamentals of
Finsler geometry, including Jacobi fields and curvature tensors,
variation formulas for arc length, and some classical comparison
theorems. Part II goes on to introduce the weighted Ricci
curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler
manifolds. These tools allow the derivation of the
Bochner-Weitzenboeck formula and the corresponding Bochner
inequality, gradient estimates, Bakry-Ledoux's Gaussian
isoperimetric inequality, and functional inequalities in the
Finsler setting. Part III comprises advanced topics: a
generalization of the classical Cheeger-Gromoll splitting theorem,
the curvature-dimension condition, and the needle decomposition.
Throughout, geometric descriptions illuminate the intuition behind
the results, while exercises provide opportunities for active
engagement. Comparison Finsler Geometry offers an ideal gateway to
the study of Finsler manifolds for graduate students and
researchers. Knowledge of differentiable manifold theory is
assumed, along with the fundamentals of functional analysis.
Familiarity with Riemannian geometry is not required, though
readers with a background in the area will find their insights are
readily transferrable.
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