Riemannian Geometry of the Free Step-Two Carnot Group

Abstract

The purpose of this paper is to give a complete description of some basic Riemannian objects on the free step-two Carnot group N_3,2. The author first finds out all the conjugate points along any geodesic starting from o by means of calculating the Jacobian determinant of the Riemannian exponential map. Next, the author determines the cut locus and the explicit expression of the squared distance d²(g). Moreover, the author obtains the optimal syntheses from o and proves that the first conjugate point coincides with the cut point.

Keywords

Free step-two Carnot group / Conjugate point / Cut locus / 22E25 / 53C22

References

[1] Li, H.-Q., The Carnot-Carathéodory distance on 2-step groups, 2021, arXiv: 2112.07822.

[2] Montanari, A. and Morbidelli, D., On the sub-Riemannian cut locus in a model of free two-step Carnot group, Calc. Var. Partial Differential Equations, 56, 2017, Paper No. 36, 26 pp.

[3] Wolf J A. Curvature in nilpotent Lie groups. Proc. Amer. Math. Soc.. 1964, 15: 271-274.

[4] Milnor J. Curvatures of left invariant metrics on Lie groups. Advances in Math.. 1976, 21: 293-329.

[5] Hadamard J. Les surfaces à courbure opposée et leurs lignes géodésiques. J. Math. Pures Appl.. 1898, 4: 27-73

[6] Cartan É. Leçons sur la Géométrie des Espaces de Riemann. 1946, Paris, Gauthier-Villars(in French)

[7] Cheeger J, Ebin D G. Comparison Theorems in Riemannian Geometry. 2008, Providence, RI, AMS Chelsea Publishing.

[8] Gulliver R. On the variety of manifolds without conjugate points. Trans. Amer. Math. Soc.. 1975, 210: 185-201.

[9] Berger M. A Panoramic View of Riemannian Geometry. 2003, Berlin, Springer-Verlag.

[10] Kaplan A. On the geometry of groups of Heisenberg type. Bull. London Math. Soc.. 1983, 15: 35-42.

[11] Kaplan A. Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata. 1981, 11: 127-136.

[12] Eberlein P. Geometry of 2-step nilpotent groups with a left invariant metric. Ann. Sci. Ecole Norm. Sup. (4). 1994, 27: 611-660.

[13] Biggs R, Nagy P T. On sub-Riemannian and Riemannian structures on the Heisenberg groups. J. Dyn. Control Syst.. 2016, 22: 563-594.

[14] Walschap G. Cut and conjugate loci in two-step nilpotent Lie groups. J. Geom. Anal.. 1997, 7: 343-355.

[15] Gornet R, Mast M B. Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds. J. Geom. Anal.. 2003, 13: 107-143.

[16] Jang C, Park K. Conjugate points on 2-step nilpotent groups. Geom. Dedicata. 2000, 79: 65-80.

[17] Li, H.-Q. and Zhang, Y., Sub-Riemannian geometry on some step-two Carnot groups, 2021, arXiv: 2102.09860.

[18] Li, H.-Q. and Zhang, Y., A complete answer to the Gaveau–Brockett problem, 2021, arXiv: 2112.07927.

[19] Calin O, Chang D-C, Furutani K, Iwasaki C. Heat Kernels for Elliptic and Sub-elliptic Operators, Methods and Techniques. 2011, New York, Birkhauser/Springer.

[20] Krantz S G, Parks H R. The Implicit Function Theorem, History, Theory, and Applications. 2002, Boston, MA, Birkhäuser Boston, Inc.

[21] Varadhan S R S. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math.. 1967, 20: 431-455.

[22] Varadhan S R S. Diffusion processes in a small time interval. Comm.. Pure Appl. Math.. 1967, 20: 659-685.

[23] Bishop R L. Decomposition of cut loci. Proc. Amer. Math. Soc.. 1977, 65: 133-136.

[24] Wolter F E. Distance function and cut loci on a complete Riemannian manifold. Arch. Math.. 1979, 32: 92-96.

[25] Sakai T. Riemannian Geometry. 1996, Providence, RI, American Mathematical Society. Translated from the 1992 Japanese original by the author149

RIGHTS & PERMISSIONS

The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg