Affine symplectic geometry. I: Applications to geometric inequalities
type: Item
instance of: scholarly article
title: Affine symplectic geometry. I: Applications to geometric inequalities
zbMATH Open document ID: 0772.53047
DOI: 10.1007/BF02808163
author: Alexander G. Reznikov
published in: Israel Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: Following the ideas of [\textit{C. B. Croke}, Ann. Sci. Éc. Norm. Super., IV. Ser. 13, 419-435 (1980; Zbl 0465.53032)] and [\textit{M. Berger}, Differ. Geom. Relativ., Vol. Honour A. Lichnerowicz 60th Birthday, 33-42 (1976; Zbl 0342.53038)], the author applies the canonical symplectic structure in the space of geodesics of a Riemannian manifold to obtain various geometric inequalities. He gives a sharp estimate from below of the area of bubble, spanned on a given curve in \(\mathbb{R}^ 3\), and investigates an `` inverse Croke inequality'' relating the area of the complex domain boundary in a Riemannian manifold to the injectivity radius and the volume of the domain. An estimate of the first eigenvalue of the Dirichlet problem (in the case of non-positive curvature) and a sharp estimate of the first eigenvalue of an operator corresponding to the harmonic oscillator on a Hadamard manifold are given.
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 219415
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: symplectic structure
zbMATH Keywords: space of geodesics
zbMATH Keywords: inverse Croke inequality
zbMATH Keywords: first eigenvalue
reviewed by: Stanisław Janeczko
MaRDI profile type: Publication
cites work: Q4167627
cites work: Q4169382
cites work: Lectures on geodesics in Riemannian geometry
cites work: Q4110284
cites work: Geometry. I, II. Transl. from the French by M. Cole and S. Levy
cites work: Spectral geometry: direct and inverse problems. With an appendix by G. Besson
cites work: Q3231775
cites work: Q4039749
cites work: A generalization of the isoperimetric inequality
cites work: A sharp four dimensional isoperimetric inequality
cites work: Some isoperimetric inequalities and eigenvalue estimates
cites work: Q4139272
cites work: The isoperimetric inequality
cites work: Some Integral Formulas for Space Curves and Their Generalization
cites work: Q3747306
cites work: Q3994028
cites work: Q4066072
cites work: Q4110292
cites work: Supersymmetry and Morse theory
description: scientific article; zbMATH DE number 219415