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 ${\displaystyle p}$ 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

• Four-vertex theorem
• Geodesics on an ellipsoid

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.