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A NOTE ON FOCAL POINT COMPARISON IN RIEMANNIAN GEOMETRY

Kim, Jong-Ryul

  • Received : 2012.04.17
  • Accepted : 2012.05.10
  • Published : 2012.06.25

Abstract

Using a comparison of the Jacobi differential equations instead of volume comparison by Itokawa in [3], the focal radius is estimated under the suitable Ricci and mean curvature conditions.

Keywords

H-Jacobi tensor;Jacobi equation;Mean curvature

References

  1. Eschenburg J.-H., O'Sullivan, J.: Jacobi tensors and Ricci curvature, Math. Ann. 252 (1980), 1-26. https://doi.org/10.1007/BF01420210
  2. G.J. Galloway: A generalization of Myers' theorem and an application to relativistic cosmology, J. Diff. Geom. 14 (1977), 105-116.
  3. Itokawa, Y.: Distance spheres and Myers-type theorems for manifolds with lower bounds on the Ricci curvature, Illinois J. Math. 34 (1990), no. 4, 693-705.
  4. Kupeli, D. N.: On conjugate and focal points in semi-Riemannian geometry, Math. Z. 198 (1988), no. 4, 569-589. https://doi.org/10.1007/BF01162874
  5. Seon-Bu Kim, Jeong-Sik Kim, Eun-Ju Lee: An Estimate of Focal Points For Spacelike Hypersurfaces, Honam Math. J. 17 (1995), 29-39.
  6. O'Neill, B.: Semi-Riemannian Geometry with applications to Relativity, Pure Appl. Math., Academic Press, New York, 1983.

Acknowledgement

Supported by : Kunsan National University