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
editReferences
edit- ^ 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
- ^ 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.
- ^ 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.
- ^ 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.
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