Abstract
Let M be a connected Riemannian manifold. The distance function d(x, y) (x, y ∈ M), which is by definition the inf limit of the lengths of the rectifiable arcs joining x to y, defines a metric compatible with the topology of M. By the Hopf-Rinow theorem, the following conditions are equivalent:
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(i)
Every geodesic can be indefinitely extended, that is, the Levi-Cività connection of M is complete (III,3.1).
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(ii)
M is complete as a metric space.
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(iii)
Every bounded set is relatively compact.
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Notes and Bibliography
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© 1998 Hindustan Book Agency
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Borel, A. (1998). Riemannian Symmetric Spaces. In: Semisimple Groups and Riemannian Symmetric Spaces. Texts and Readings in Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-92-2_4
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DOI: https://doi.org/10.1007/978-93-80250-92-2_4
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