Abstract
Following the transition presented in chapter 8, it is now quite natural to study the geodesic behaviour of a Riemannian manifold. For local metric geometry it is natural because geodesics are locally the shortest paths. This point of view was treated in §6.5. But geodesics of any length are of interest for the geometer. Another strong motivation for the study of geodesic dynamics comes from mechanics. Since Riemannian manifolds provide a very general setting for Hamiltonian mechanics, with their geodesics being the desired Hamiltonian trajectories, we are of course interested in their behaviour for any interval of time (any length). This perspective mixes dynamics and geometry and is extremely popular today. People always want to predict the future, more or less exactly. We will comment more on this below. Dynamics plays an ever larger role in geometry, even in very simple contexts such as the study of pentagons and Pappus theorems (see Schwartz 1993,1998 [1114, 1115] and the important work d’Ambra & Gromov 1991 [426]. Note also that Gromov 1987 [622] introduced dynamical systems into the study of discrete groups.
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See page 359 for a precise definition of the geodesic flow.
Nondegeneracy of a quadratic form Q means that if we write Q(v +w) = Q(v)+ Q(w) + P(v,w) with P(v,w) = P(w, v), then there is no v for which every w satisfies P(v,w) = 0
For a change, and simplification of notation, we will now use the discrete language, leaving to the reader to translate it into the continuous one.
Curvature is always understood here to mean sectional curvature, unless otherwise noted.
Topological transitivity means the existence of at least one trajectory which is everywhere dense
See page 248 for the definition of Jacobi field.
See the definition of Busemann function in definition 334 on page 585.
This UM is the unit tangent bundle of the universal covering M of M.
References are the same as the ones at the beginning of §§§10.5.1.1 and add Pansu 1991 [996] for a fast informative survey, and Knieper 1998 [821].
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© 2003 Springer-Verlag Berlin Heidelberg
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Berger, M. (2003). Riemannian Manifolds as Dynamical Systems: the Geodesic Flow and Periodic Geodesics. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_10
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DOI: https://doi.org/10.1007/978-3-642-18245-7_10
Publisher Name: Springer, Berlin, Heidelberg
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