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Egloff, Daniel

Uniform Finsler Hadamard manifolds. (English) Zbl 0919.53020

Ann. Inst. Henri Poincaré, Phys. Théor. 66, No. 3, 323-357 (1997).
Given a Finsler manifold \(M\), the fundamental tensor gives rise to a family of direction-dependent inner products on each tangent space. Each such inner product induces a norm on the tangent space in question. If the norms induced by these inner products are all equivalent to each other through a single constant which is usable for all tangent spaces, the Finsler manifold in question is said to be uniform.
The present paper studies Finsler manifolds which are uniform, reversible (that is, absolutely homogeneous), simply connected, complete, and which have nonpositive flag curvature. It contains a careful discussion of the exponential map, asymptotic geodesics and the visual boundary, visibility and \(\delta\)-hyperbolicity, as well as Hilbert geometries. It proves that if the nonpositive flag curvature is uniformly bounded away from \(0\), then these uniform Finsler Hadamard manifolds are uniformly visible, and are hyperbolic in the sense of Gromov.
Readers who plan to understand this subject in greater details may also want to consult the author’s dissertation.
Reviewer: David Bao (Houston)

Cited in 19 Documents

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

uniform Finsler manifold; nonpositive flag curvature; asymptotic geodesics; visual boundary; visibility; \(\delta\)-hyperbolicity; Hilbert geometries; Finsler Hadamard manifolds
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\textit{D. Egloff}, Ann. Inst. Henri Poincaré, Phys. Théor. 66, No. 3, 323--357 (1997; Zbl 0919.53020)
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