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International youth conference "Geometry & Control"
April 17, 2014 16:05, Moscow, Steklov Mathematical Institute of RAS
 


Local Conformal Flatness of Left-Invariant 3D Contact Structures

Francesco Boarotto

SISSA, International School for Advanced Studies, Trieste, Italy
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Francesco Boarotto


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Abstract: In this talk I want to address the problem of finding the locally flat left-invariant contact structures on a three dimensional Lie Group up to conformal transformations, that is I will determine the ones locally conformally equivalent to the Heisenberg algebra $\mathbb H_3$. In particular I will show how to build the Fefferman metric associated to a generic three dimensional contact structure (not necessarily left-invariant) and by means of this construction I will give the explicit formula for the (unique) conformal invariant associated to such a structure. Next, specializing the study to the left-invariant case, I will give a complete list of the locally conformally flat structures which may appear and I will find the explicit form of the maps $\varphi: M\to \mathbb R$ which flatten our structures, and I will show that they are essentially (i.e. up to multiplication by a constant) unique.
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Theorem. Let $(M,\Delta,g)$ be a left-invariant 3D contact structure. Then it is locally conformally flat if and only if its canonical frame satisfies one of the following
\begin{equation*} i)\; \{ \begin{array}{lll} [f_2,f_1]&=&f_0+c_{12}^2f_2,



[f_1,f_0]&=&\frac29(c_{12}^2)^2f_2,



[f_2,f_0]&=&0.
\end{array} .\qquadii)\; \{ \begin{array}{lll} [f_2,f_1]&=&f_0+c_{12}^1f_1,



[f_1,f_2]&=&0,



[f_2,f_0]&=&-\frac29(c_{12}^1)^2f_2.
\end{array} . \end{equation*}
or
\begin{equation*} iii)\; \{ \begin{array}{lll} [f_2,f_1]&=&f_0,

[f_1,f_0]&=&\kappa f_2,

[f_2,f_0]&=&-\kappa f_1,\qquad \kappa<0. \end{array} . \end{equation*}
Where $\kappa$ is the curvature of the structure.
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Open question 1. Give a complete classification (i.e. not just the locally conformally flat ones) of left-invariant three dimesional contact structures, up to real rescalings.
$ $
Open question 2. Give satisfactory criteria to determine whether a given three dimesional contact structure (not necessarily left-invariant) is locally conformally flat or not.

Materials: abstract.pdf (70.2 Kb)

Language: English

References
  1. A. A. Agrachev, “Exponential mappings for contact sub-Riemannian structures”, J. Dynamical and Control Systems, 2 (1996), 321–358  crossref  mathscinet  zmath  scopus
  2. A. A. Agrachev, D. Barilari, “Sub-Riemannian structures on 3D Lie groups”, J. Dynamical and Control Systems, 18 (2012), 21–44  crossref  mathscinet  zmath  isi  scopus
  3. A. L. Castro, R. Montgomery, “The chains of left-invariant Cauchy-Riemann structures on $SU(2)$”, Pacific J. Math, 238:1 (2008), 41–71  crossref  mathscinet  zmath  isi  scopus
  4. F. A. Farris, “An intrinsic construction of Fefferman's CR metric”, Pacific J. Math, 123:1 (1986), 33–45  crossref  mathscinet  zmath  isi
  5. C. Fefferman, C. R. Graham, “The ambient metric”, Annals of mathematics studies, 178, Princeton University press, NJ, 2012, x+113. pp.  mathscinet  zmath
  6. J. M. Lee, “The Fefferman metric and pseudo-Hermitian invariants”, Trans. Amer. Math. Soc., 296:1 (1986), 411–429  mathscinet  zmath  isi


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