scholarly journals Moduli of Germs of Legendrian Curves

2009 ◽  
Vol 18 (4) ◽  
pp. 797-809
Author(s):  
António Araújo ◽  
Orlando Neto
Keyword(s):  
2017 ◽  
Vol 153 (9) ◽  
pp. 1945-1986 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič ◽  
Francisco J. López

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on$\mathbb{C}^{2n+1}$for any$n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface$M$admits a proper holomorphic Legendrian embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface$M={M\unicode[STIX]{x0030A}}\,\cup \,bM$there exists a topological embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$whose restriction to the interior is a complete holomorphic Legendrian embedding${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold$W$carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on$W$.


2002 ◽  
Vol 11 (07) ◽  
pp. 1077-1087
Author(s):  
MARCOS M. DINIZ

The formula Lk = Wr + Tw, which expresses the linking number of two curves that bound a ribbon as a sum of two terms, has particularly interested biologists and was used to understand the DNA structure. The study of Legendrian curves in contact manifolds, and in particular in the Heisenberg space, is attached to some important problems in geometry, as the problem of classification of contact structures. In this work, we show the analogue formula for curves in the Heisenberg space, we relate the writhing number with the Thurston-Benequin invariant of a Legendrian curve and derive some results directly from this formula.


2017 ◽  
Vol 2019 (3) ◽  
pp. 893-922 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič
Keyword(s):  

2017 ◽  
Vol 288 (1-2) ◽  
pp. 643-663
Author(s):  
Franc Forstnerič ◽  
Finnur Lárusson

1996 ◽  
Vol 05 (06) ◽  
pp. 743-778 ◽  
Author(s):  
FRANCESCA AICARDI

An invariant polynomial s(t) is defined for framed knots in the solid torus. The coefficients are Vassiliev invariants of order one. An invariant polynomial A(t) of Legendrian curves is introduced and it is shown how to calculate it from their fronts. The coefficient of A(t) of the order n term is the restriction to the discriminant of the selftangencies with partial index n of the Arnold invariant J+ of wave fronts. The polynomial A(t) of a Legendrian curve is recovered from the polynomial s(t) of the Legendrian knot, provided with its natural contact framing.


2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Ali Maalaoui ◽  
Vittorio Martino

AbstractIn this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S


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