How Do Nearby Geodesics Deviate in Curved Spaces?

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SUMMARY

The discussion centers on the Jacobi geodesic deviation formula, which mathematically characterizes how geodesics deviate in curved spaces using the Riemannian curvature tensor. This formula is applicable in both Riemannian and Lorentzian geometries, making it a fundamental concept in understanding geodesic behavior. Participants seek resources that provide clarity on the geodesic deviation equation, particularly in contexts outside of relativity.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with the Riemannian curvature tensor
  • Basic knowledge of geodesics
  • Concepts of Lorentzian geometry
NEXT STEPS
  • Study the Jacobi geodesic deviation formula in detail
  • Explore textbooks on Riemannian geometry for comprehensive explanations
  • Investigate the applications of the Riemannian curvature tensor
  • Learn about the differences between Riemannian and Lorentzian geometries
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Mathematicians, physicists, and students studying differential geometry or general relativity who seek to understand the behavior of geodesics in curved spaces.

ledol83
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Hi...does anyone have a good description (or a link to it) on geodesic deviation equation?Most of the references i have are in a setting of relativity, which make me all at sea.

Please help me if you know a mathematical characterization of how geodesics from one point deviate (which just involves the Riemannian curvature tensor).

Thanks a bunch!
 
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That's the Jacobi geodesic deviation formula, which is discussed in many textbooks on Riemannian geometry (the Lorentzian version is almost identical).
 

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