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Comparison theorems for volumes in surfaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1167)

Abstract

Let M and \(\tilde M\)be complete surfaces such that the respective sectional curvatures satisfy \(K \geqslant \widetilde{\rm K}\). It is shown that the volumes of geodesic balls satisfy \(V_m (r) \leqslant \tilde V_{\tilde m} (r)\). If M and \(\tilde M\)are compact and simply connected then vol(M)≤vol(\(\tilde M\)).

Keywords

  • Vector Bundle
  • Sectional Curvature
  • Gaussian Curvature
  • Riccati Equation
  • Comparison Theorem

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References

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© 1985 Springer-Verlag

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Gray, A. (1985). Comparison theorems for volumes in surfaces. In: Geometry and Topology. Lecture Notes in Mathematics, vol 1167. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075219

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  • DOI: https://doi.org/10.1007/BFb0075219

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16053-3

  • Online ISBN: 978-3-540-39738-0

  • eBook Packages: Springer Book Archive