Advertisement

manuscripta mathematica

, Volume 68, Issue 1, pp 209–214 | Cite as

Comparison theory for Riccati equations

  • J. -H. Eschenburg
  • E. Heintze
Article

Abstract

We give a short new proof for the comparison theory of the matrix valued Riccati equationB′+B 2+R=0 with singular initial values. Applications to Riemannian geometry are briefly indicated.

Keywords

Riccati Equation Comparison Theorem Positive Semidefinite Shape Operator Comparison Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Eb] P. Eberlein:When is a geodesic flow Anosov? I. J. Differential Geometry8, 437–463 (1973)zbMATHMathSciNetGoogle Scholar
  2. [E1] J.-H. Eschenburg:Stabilitätsverhalten des Geodätischen Flusses Riemannscher Mannigfaltigkeiten. Bonner Math. Schr.87 (1976)Google Scholar
  3. [E2] J.-H. Eschenburg:Comparison theorems and hypersurfaces. Manuscripta math.59, 295–323 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [Gr] A. Gray:Comparison Theorems for the volume of tubes as generalizations of the Weyl tube formula. Topology21, 201–228 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Gn] L.W. Green:A theorem of E. Hopf. Mich. Math. J.5, 31–34 (1958)zbMATHCrossRefGoogle Scholar
  6. [GM] D. Gromoll & W. Meyer:Examples of complete manifolds with positive Ricci curvature. J. Differential Geometry21, 195–211 (1985)zbMATHMathSciNetGoogle Scholar
  7. [Gv] M. Gromov:Structures metriques pour les variétés riemanniennes. Cedic-Nathan 1981Google Scholar
  8. [HE] S.W. Hawking, G.F.R.Ellis:The Large Scale Structure of Spacetime. Cambridge 1973Google Scholar
  9. [HIH] E. Heintze & H.C. Im Hof:Geometry of horospheres. J. Differential Geometry12, 481–491 (1977)zbMATHMathSciNetGoogle Scholar
  10. [K1] H. Karcher:Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math.30, 509–541 (1977)zbMATHMathSciNetGoogle Scholar
  11. [K2] H. Karcher:Riemannian Comparison Constructions. S.S. Chern (ed.) Global Differential Geometry, M.A.A. Studies in Mathematics vol.27 Google Scholar
  12. [R] W.T. Reid:Riccati Differential equations. Academic Press 1972Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • J. -H. Eschenburg
    • 1
  • E. Heintze
    • 1
  1. 1.Institut für MathematikAugsburg

Personalised recommendations