A Cartan-Hadamard theorem for Banach-Finsler manifolds
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 accounts of the results so far obtained in this study, the author proves an existence theorem for a symmetric Banach-Lie group with an involution to have a polar decomposition. Finally, typical examples of symmetric Finsler manifolds with seminegative curvature are provided which are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure but, in general, are not Riemannian. Moreover, some open problems arising in the context of this study, are given. The text of this paper is nicely organized, the paper is very well written and is highly interesting.
scientific article; zbMATH DE number 1879689