Last geometric statement of Jacobi

Summary

In differential geometry the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi. According to this conjecture:

Every caustic from any point on an ellipsoid other than umbilical points has exactly four cusps.[1]

While numerical experiments had indicated the statement is true,[2] it wasn’t until 2004 that it was proven rigorously by Itoh and Kiyohara.[3]

It has since been extended to a wider class of surfaces beyond the ellipsoid.[4]

References

  1. ^ Arnold, V. I. (1999), "Topological problems in wave propagation theory and topological economy principle in algebraic geometry", The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., vol. 24, Providence, RI: Amer. Math. Soc., pp. 39–54, MR 1733567
  2. ^ Sinclair, R. (2003). "On the last geometric statement of Jacobi". Experimental Mathematics. 12 (4): 477–485. doi:10.1080/10586458.2003.10504515. MR 2043997.
  3. ^ Itoh, J.; Kiyohara, K. (2004). "The cut loci and the conjugate loci on ellipsoids". Manuscripta Mathematica. 114 (2): 247–264. doi:10.1007/s00229-004-0455-z. S2CID 121131543.
  4. ^ Sinclair, R.; Tanaka, M. (2006). "Jacobi's last geometric statement extends to a wider class of Liouville surfaces". Mathematics of Computation. 75 (256): 1779–1808. doi:10.1090/S0025-5718-06-01924-7. MR 2240635.