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We have already mentioned extensions of various results for compact manifolds to complete manifolds. It is clear that one should restrict oneself to only certain sorts of noncompact Riemannian manifolds to have a hope of obtaining results. Let us mention in particular the possibilities of examining manifolds with finite volume, those with prescribed asymptotic behaviour at infinity, for example quadratic decay,1 quadratic curvature decay, volume behaviour, Euclidean asymptoticity, etc.

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  1. Quadratic decay means that the curvature decreases as one goes to infinity, at least as r −2 where r is the distance to an arbitrary chosen point; there are variations on this definition.

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  2. For further remarks, added 2007, see page 723

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  3. Obvious definition: one demands that at the boundary the manifold is smoothly equivalent to one side of an hyperplane in ℝd, so that the boundary is smooth.

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© 2003 Springer-Verlag Berlin Heidelberg

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Berger, M. (2003). Some Other Important Topics. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65317-2

  • Online ISBN: 978-3-642-18245-7

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