Poincaré’s Lemma on Some Non-Euclidean Structures

Abstract

The author proves the Poincaré lemma on some (n + 1)-dimensional corank 1 sub-Riemannian structures, formulating the $\frac{{\left( {n - 1} \right)n\left( {{n^2} + 3n - 2} \right)}}{8}$ necessarily and sufficiently “curl-vanishing” compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.

Keywords

Poincaré lemma / Cesàro-Volterra path integral / Sub-Riemannian manifolds

References

[1] Abraham R., Marsden J. E., Ratiu T.. Manifolds, Tensor Analysis, and Applications, 1988 2 New York: Springer-Verlag

[2] Agrachev A., Barilari D., Boscain U.. Introduction to Geodesics in Sub-Riemannian Geometry, Geometry Analysis and Dynamics on Sub-Kiemannian Manifolds, 2016, Zürich: EMS Ser. Lect. Math., Eur. Math. Soc.

[3] Amrouche Ch., Ciarlet P. G., Mardare C.. Remarks on a lemma by Jacques-Louis Lions. C. R. Math. Acad. Sci. Paris, 2014, 352(9): 691-695

[4] Amrouche Ch., Ciarlet P. G., Mardare C.. On a lemma of Jacques-Louis Lions and its relation to other fundamental results. J. Math. Pures Appl. (9), 2015, 104(2): 207-226

[5] Calin O., Chang D.-C.. Sub-Riemannian Geometry, General Theory and Examples, Encyclopedia Math. Appl., 2009, Cambridge: Cambridge University Press

[6] Calin O., Chang D.-C., Eastwood M.. Integrability conditions for the Grushin and Martinet distributions. Bull. Inst. Math. Acad. Sin. (N.S.), 2013, 8(2): 159-168

[7] Calin O., Chang D.-C., Eastwood M.. Integrability conditions for Heisenberg and Grushin-type distributions. Anal. Math. Phys., 2014, 4(1–2): 99-114

[8] Calin O., Chang D.-C., Hu J.. Poincaré’s lemma on the Heisenberg group. Adv. in Appl. Math., 2014, 60: 90-102

[9] Calin O., Chang D.-C., Hu J.. Integrability conditions on Engel-type manifolds. Anal. Math. Phys., 2015, 5(3): 217-231

[10] Calin O., Chang D.-C., Hu J.. Integrability conditions on a sub-Riemannian structure on S³. Anal. Math. Phys., 2017, 7(1): 9-18

[11] Chen W., Jost J.. A Riemannian version of Korn’s inequality. Calc. Var. Partial Differential Equations, 2012, 14: 517-530

[12] Ciarlet P. G.. Linear and nonlinear functional analysis with applications, 2013, Philadelphia, PA: Society for Industrial and Applied Mathematics

[13] Ciarlet P. G., Gratie L., Mardare C., Shen M.. Saint Venant compatibility equations on a surface application to intrinsic shell theory. Math. Models Methods Appl. Sci., 2008, 18(2): 165-194

[14] Ciarlet P. G., Mardare S.. Nonlinear Saint-Venant compatibility conditions and the intrinsic approach for nonlinearly elastic plates. Math. Models Methods Appl. Sci., 2013, 23(12): 2293-2321

[15] Ciarlet P. G., Mardare S., Shen M.. Saint Venant compatibility equations in curvilinear coordinates. Anal. Appl. (Singap.), 2007, 5(3): 231-251

[16] do Carmo M. P.. Riemannian Geometry, 1992, Boston: Birkhäuser

[17] Figalli A., Rifford L.. Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal., 2010, 20(1): 124-159

[18] Ohta S., Sturm K.-T.. Heat flow on Finsler manifolds. Comm. Pure Appl. Math., 2009, 62(10): 1386-1433

[19] Rizzi L.. Measure contraction properties of Carnot groups. Calc. Var. Partial Differential Equations, 2016