Last geometric statement of Jacobi

In differential geometry the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi, which states:

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

Numerical experiments had indicated the statement is true[2] before it was proven rigorously in 2004 by Itoh and Kiyohara.[3] It has since been extended to a wider class of surfaces beyond the ellipsoid.[4]

See also

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. S2CID 13520470.
  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. Bibcode:2006MaCom..75.1779S. doi:10.1090/S0025-5718-06-01924-7. MR 2240635.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.