A Cartan-Hadamard theorem for Banach-Finsler manifolds
type: Item
instance of: scholarly article
title: A Cartan-Hadamard theorem for Banach-Finsler manifolds
zbMATH Open document ID: 1027.58003
DOI: 10.1023/A:1021221029301
published in: Geometriae Dedicata
publication date: 2003-03-11T00:00:00Z
review text: This paper is concerned with the studies of a Banach-Finsler manifold \(( M, b, F)\) modelled over the Banach manifold \(M\), equipped with a compatible tangent norm \(b\) and a spray \(F\), which have a seminegative curvature in the sense that corresponding exponential map has a surjective expansive differential in every point [see \textit{S. Lang}, Fundamentals of Differential Geometry, Springer-Verlag, New York (1999; Zbl 0932.53001)]. The author first generalizes a classical theorem of Cartan-Hadamard, stating that the exponential map is a covering map, to the setting of Banach-Finsler manifold. A geometric key result of section 1 is the theorem: ``Let \( (X, b_{X}) \) be a complete Finsler manifold, \((Y,b_{Y},F)\) be a connected Finsler manifold with spray and let \( f:X \to Y\) be a local \( C^{1}\) -diffeomorphism for which there exists a constant \(C > 0 \) such for all \( w \in TX \), we have \( b_{Y} (Tf(w))\geq C.b_{X}(w)\). Then \(f\) is surjective, \(f\) is a covering and has the unique path lifting property, and \(Y\) is complete.'' This is, indeed, the generalization of a theorem [S. Lang (loc. cit.), VIII] from a Riemannian manifold to a Finsler manifold. Next, the author discusses some basic facts about bounded dissipative especially Hermitian operators on Banach spaces and then turn to the studies of symmetric spaces. If the symmetric space \(M\) can be written as \(G/K\), where \(G\) is a Banach-Lie group and \(K\) an open subgroup of the group of fixed points of an involution \(\sigma \), then one can derive a criterion for a \(G\)-invariant Finsler metric on \(M\) to lead to a manifold with seminegative curvature. Using the results from section 2, it is shown that \(M\) has seminegative curvature if the operators \(-(a d/ x)^{2}/_{P}, x\in P, P \cong T_0(M)\), are dissipative. Thereafter, a criterion is elaborated for symmetric Banach-Lie algebras which makes it easier to check the condition derived in the previous section. Subsequently, taking…
reviewed by: Om Prakash Singh
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C60
zbMATH DE Number: 1879689
zbMATH Keywords: symmetric space
zbMATH Keywords: Banach manifold
zbMATH Keywords: tangent norm
zbMATH Keywords: Finsler metric
zbMATH Keywords: Cartan-Hadamard Theorem
author: Karl-Hermann Neeb
MaRDI profile type: Publication
full work available at URL: a:1021221029301
OpenAlex ID: W2353214541
description: scientific article; zbMATH DE number 1879689