scholarly journals Moduli of Germs of Legendrian Curves

2009 ◽  
Vol 18 (4) ◽  
pp. 797-809
Author(s):  
António Araújo ◽  
Orlando Neto
Keyword(s):  
Legendrian Curves

scholarly journals Holomorphic Legendrian curves

2017 ◽  
Vol 153 (9) ◽  
pp. 1945-1986 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič ◽  
Francisco J. López
Keyword(s):  
Riemann Surface ◽  
Contact Structure ◽  
Existence Theorems ◽  
Contact Manifold ◽  
Open Problems ◽  
General Existence ◽  
The Subject ◽  
Complex Contact Manifold ◽  
Holomorphic Contact ◽  
Legendrian Curves

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$W$.

A SELF-LINKING FORMULA FOR CURVES IN THE HEISENBERG SPACE

2002 ◽  
Vol 11 (07) ◽  
pp. 1077-1087
Author(s):  
MARCOS M. DINIZ
Keyword(s):  
Dna Structure ◽  
Contact Structures ◽  
Contact Manifolds ◽  
Linking Number ◽  
Legendrian Curves

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.

scholarly journals Darboux Charts Around Holomorphic Legendrian Curves and Applications

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

scholarly journals The Oka principle for holomorphic Legendrian curves in $$\mathbb {C}^{2n+1}$$ C 2 n + 1

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

INVARIANT POLYNOMIAL OF FRAMED KNOTS IN THE SOLID TORUS AND ITS APPLICATION TO WAVE FRONTS AND LEGENDRIAN KNOTS

1996 ◽  
Vol 05 (06) ◽  
pp. 743-778 ◽  
Author(s):  
FRANCESCA AICARDI
Keyword(s):  
Solid Torus ◽  
Invariant Polynomial ◽  
Partial Index ◽  
Legendrian Knots ◽  
Wave Fronts ◽  
Vassiliev Invariants ◽  
Legendrian Knot ◽  
Legendrian Curves

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.

scholarly journals The Topology of a Subspace of the Legendrian Curves on a Closed Contact 3-Manifold

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Ali Maalaoui ◽  
Vittorio Martino
Keyword(s):  
Loop Space ◽  
Closed Contact ◽  
Legendrian Curves

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

Sign in / Sign up

Export Citation Format

Share Document