On the Cauchy - Riemann Geometry of Transversal Curves in the 3-Sphere
DOI:
https://doi.org/10.15407/mag16.03.312Анотація
Нехай $S^3$ є одиничною сферою у $\mathbb{C}^2$ зі стандартною структурою Коші-Рімана (CR). Використовуючи локальні CR інваріанти $S^3$, у цій статті досліджено CR геометрію кривих в $S^3$, які трансверсальні до контактного розподілу. А саме, у центрі уваги є CR геометрія трансверсальних вузлів. Розглянуто чотири глобальні інваріанти трансверсальних вузлів: фазова аномалія, CR спін, індекс Маслова та CR число самозацеплення. Обговорюється зв’язок між цими інваріантами і числом Беннеквіна вузла. Також розглянуто найпростішу CR інваріантну варіаційну проблему для загальних трансверсальних кривих і досліджено замкнуті критичні криві.Mathematics Subject Classification: 53C50, 53C42, 53A10
Ключові слова:
CR геометрія тривимірної сфери, контактна геометрія, трансверсальні криві, CR інваріанти трансверсальних вузлів, число самозацеплення, число Беннеквіна, деформація функціоналу для трансверсальних кривих, критичні вузлиПосилання
T.F. Banchoff, Osculating tubes and self-linking for curves on the three-sphere, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001, 10–19. https://doi.org/10.1090/conm/288/04813
E. Barletta and S. Dragomir, Robinson–Sparling construction of CR structures associated to shearfree null geodesic congruences, Riv. Mat. Univ. Parma 11 (2020), to appear.
D. Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, 1 (Schnepfenried, 1982), Astérisque, 107–108, Soc. Math. France, Paris, 1983, 87–161 (French).
D. Burns, Jr., K. Diederich, and S. Shnider, Distinguished curves in pseudoconvex boundaries, Duke Math. J. 44 (1977), No. 2, 407–431. https://doi.org/10.1215/S0012-7094-77-04419-2
R.L. Bryant, On notions of equivalence of variational problems with one independent variable, Contemp. Math. 68 (1987), 65–76. https://doi.org/10.1090/conm/068/924805
G. Călugăreanu, L’intégral de Gauss et l’analyse des noeuds tridimensionnels, Rev. Math. Pures Appl. 4 (1959), 5–20 (French).
E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I, Ann. Math. Pura Appl. (4) 11 (1932), 17–90; or Oeuvres II, 2, 1931– 1304 (French). https://doi.org/10.1007/BF02417822
E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, II, Ann. Scuola Norm. Sup. Pisa (2) 1 (1932), 333–354; or Oeuvres III, 2, 1217–1238 (French).
Y. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002), No. 3, 441–483. https://doi.org/10.1007/s002220200212
S.S. Chern and R.S. Hamilton, On Riemannian metrics adapted to threedimensional contact manifolds. With an appendix by Alan Weinstein, Lecture Notes in Math., 1111, Workshop Bonn 1984 (Bonn, 1984), Springer, Berlin, 1985, 279– 308. https://doi.org/10.1007/BFb0084596
S.S. Chern and J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. https://doi.org/10.1007/BF02392146
D. DeTurck and H. Gluck, Linking, twisting, writhing, and helicity on the 3-sphere and in hyperbolic 3-space, J. Differential Geom. 94 (2013), No. 1, 87–128. https://doi.org/10.4310/jdg/1361889062
A. Dzhalilov, E. Musso and L. Nicolodi, Conformal geometry of timelike curves in the (1+2)-Einstein universe, Nonlinear Anal. 143 (2016), 224–255. https://doi.org/10.1016/j.na.2016.05.011
Y. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 171–193.
J.B. Etnyre, Transversal torus knots, Geom. Topol. 3 (1999), 253–268. https://doi.org/10.2140/gt.1999.3.253
J.B. Etnyre, Legendrian and transveral knots, Hanbook of Knot Theory (Eds. W. Menasco and M. Thistlethwaite), Elsevier B. V., Amsterdam, 2005, 105–185; or https://arxiv.org/abs/math/0306256v2. https://doi.org/10.1016/B978-044451452-3/50004-6
J.B. Etnyre and K. Honda, Knots and contact geometry I: torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), 63–120. https://doi.org/10.4310/JSG.2001.v1.n1.a3
F.A. Farris, An intrinsic construction of Fefferman’s CR metric, Pacific J. Math. 123 (1986), No. 1, 33–45. https://doi.org/10.2140/pjm.1986.123.33
C.L. Fefferman, Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), No. 2, 395–416; correction: 104 (1976), 393–394. https://doi.org/10.2307/1970945
D. Fuchs and S. Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), No. 5, 1025–1053. https://doi.org/10.1016/S0040-9383(96)00035-3
F.B. Fuller, The writhing number of a space curve, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 815–819. https://doi.org/10.1073/pnas.68.4.815
H. Gluck and L.-H. Pan, Embedding and knotting of positive curvature surfaces in 3-space, Topology 37 (1998), No. 4, 851–873. https://doi.org/10.1016/S0040-9383(97)00056-6
J.D. Grant and E. Musso, Coisotropic variational problems, J. Geom. Phys. 50 (2004), 303–338. https://doi.org/10.1016/j.geomphys.2003.10.005
P.A. Griffiths, Exterior differential systems and the calculus of variations, Progress in Mathematics, 25, Birkhäuser, Boston, 1982. https://doi.org/10.1007/978-1-4615-8166-6
L. Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom. 36 (1992), 551–589. https://doi.org/10.4310/jdg/1214453181
H. Jacobowitz, Chains in CR geometry, J. Differential Geom. 21 (1985), No. 2, 163–194. https://doi.org/10.4310/jdg/1214439561
G.R. Jensen, E. Musso, and L. Nicolodi, The geometric Cauchy problem for the membrane shape equation, J. Phys. A 47 (2014), No. 49, 495201-22. https://doi.org/10.1088/1751-8113/47/49/495201
G. R. Jensen, E. Musso, and L. Nicolodi, Surfaces in Classical Geometries. A Treatment by Moving Frames, Universitext, Springer, Cham, 2016. https://doi.org/10.1007/978-3-319-27076-0
L.K. Koch, Chains on CR manifolds and Lorentz geometry, Trans. Amer. Math. Soc. 307 (1988), No. 2, 827–841. https://doi.org/10.1090/S0002-9947-1988-0940230-2
J.M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. Amer. Math. Soc. 296 (1986), No. 1, 411–429. https://doi.org/10.2307/2000582
E.E. Levi, Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse, Ann. Mat. Pura Appl. 17 (1910), No. 1, p. 61–87 (Italian). https://doi.org/10.1007/BF02419336
E.E. Levi, Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse, Ann. Mat. Pura Appl. 18 (1911), No. 1, p. 69–79 (Italian). https://doi.org/10.1007/BF02420535
H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155–158. https://doi.org/10.2307/1970121
J. Martinet, Formes de contact sur les variétés de dimension 3, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., 209, Springer, Berlin, 1971, 142–163. https://doi.org/10.1007/BFb0068901
E. Musso, The local embedding problem for optical structures, J. Geom. Phys. 10 (1992), No. 1, 1–18. https://doi.org/10.1016/0393-0440(92)90004-K
E. Musso and L. Nicolodi, Closed trajectories of a particle model on null curves in anti-de Sitter 3-space, Classical Quantum Gravity 24 (2007), No. 22, 5401–5411. https://doi.org/10.1088/0264-9381/24/22/005
E. Musso and L. Nicolodi, Reduction for constrained variational problems on 3dimensional null curves, SIAM J. Control Optim. 47 (2008), No. 3, 1399–1414. https://doi.org/10.1137/070686470
E. Musso and L. Nicolodi, Quantization of the conformal arclength functional on space curves, Comm. Anal. Geom. 25 (2017), No. 1, 209–242. https://doi.org/10.4310/CAG.2017.v25.n1.a7
E. Musso and F. Salis, The Cauchy–Riemann strain functional for Legendrian curves in the 3-sphere, Annali di Matematica (2020), doi: https://doi.org/10.1007/ s10231-020-00974-7; or https://arxiv.org/abs/2003.01713v1.
J. O’Hara, Energy of Knots and Conformal Geometry, Series on Knots and Everything, 33, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. https://doi.org/10.1142/5229
W.F. Pohl, The self-linking number of a closed space curve, J. Math. Mech. 17 (1968), 975–985.
H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo (1907), 185–220 (French). https://doi.org/10.1007/BF03013518
I. Robinson and A. Trautman, Integrable optical geometry, Lett. Math. Phys. 10 (1985), No. 2-3, 179–182. https://doi.org/10.1007/BF00398155
I. Robinson and A. Trautman, Cauchy–Riemann structures in optical geometry, Proceedings of the fourth Marcel Grossmann meeting on general relativity, Parts A, B (Rome, 1985), North-Holland, Amsterdam, 1986, 317–324.
N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962), 397–429. https://doi.org/10.2969/jmsj/01440397
J. White, Self-linking and the Gauss integral in higher dimensions, Amer. J. Math. 91 (1969), No. 3, 693–728. https://doi.org/10.2307
