# American Institute of Mathematical Sciences

March  2021, 13(1): 145-166. doi: 10.3934/jgm.2020016

## A unifying approach for rolling symmetric spaces

 1 Wydziaƚ Matematyczno-Przyrodniczy, Uniwersytet Kardynaƚa Stefana Wy-szyńskie-go w Warszawie, ul. Dewajtis 5, 01-815 Warszawa, Poland 2 Department of Mathematics, University of Trás-os-Montes e Alto Douro (UTAD), 5001-801 Vila Real, Portugal, and, Institute of Systems and Robotics, University of Coimbra - Pólo II, 3030-290 Coimbra, Portugal 3 Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal, and, Institute of Systems and Robotics, University of Coimbra - Pólo II, 3030-290 Coimbra, Portugal

* Corresponding author: fleite@mat.uc.pt

Received  December 2019 Revised  March 2020 Published  March 2021 Early access  June 2020

Fund Project: The second and third authors acknowledge Fundaç˜ao para a Ciência e Tecnologia (FCT) and COMPETE 2020 program for the financial support to the project UIDB/00048/2020

The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces. Rolling motions provide interesting examples of nonholonomic systems and symmetric spaces appear associated to important applications. We make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This emphasises the relevance of Lie theory in the geometry of rolling manifolds and explains why many particular examples scattered through the existing literature always show a common pattern.

Citation: Krzysztof A. Krakowski, Luís Machado, Fátima Silva Leite. A unifying approach for rolling symmetric spaces. Journal of Geometric Mechanics, 2021, 13 (1) : 145-166. doi: 10.3934/jgm.2020016
##### References:
 [1] A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [2] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Springer-Verlag, New York, 2003. doi: 10.1115/1.1641775. [3] A. M. Bloch, R. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces, Comm. Math. Phys., 187 (1997), 357-373.  doi: 10.1007/s002200050140. [4] A. M. Bloch, M. Camarinha and L. Colombo, Variational point-obstacle avoidance on Riemannian manifolds, preprint, arXiv: 1909.12321[eess.SY]. [5] A. M. Bloch and P. E. Crouch, Optimal control, optimization, and analytical mechanics, in Mathematical Control Theory, Springer, New York, 1999,268–321. doi: 10.1007/978-1-4612-1416-8_8. [6] A. M. Bloch, P. E. Crouch and T. S. Ratiu, Sub-Riemannian optimal control problems, in Hamiltonian and Gradient Flows, Algorithms and Control, Fields Inst. Commun., 3, Amer. Math. Soc., Providence, RI, 1994, 35–48. [7] A. M. Bloch and A. G. Rojo, Kinematics of the rolling sphere and quantum spin, Commun. Inf. Syst., 10 (2010), 221-238.  doi: 10.4310/CIS.2010.v10.n4.a4. [8] R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math., 114 (1993), 435-461.  doi: 10.1007/BF01232676. [9] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [10] G. S. Chirikjian, Information theory on Lie groups and mobile robotics applications, IEEE International Conference on Robotics and Automation, Anchorage, AK, 2010, 2751–2757. doi: 10.1109/ROBOT.2010.5509791. [11] Y. Chitour, M. Godoy Molina and P. Kokkonen, The rolling problem: overview and challenges, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser., 5, Springer, Cham, 2014,103–122. doi: 10.1007/978-3-319-02132-4_7. [12] Y. Chitour and P. Kokkonen, Rolling manifolds on space forms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 927-954.  doi: 10.1016/j.anihpc.2012.05.005. [13] P. E. Crouch and F. Silva Leite, Rolling motions of pseudo-orthogonal groups, IEEE 51st Annual Conference on Decision and Control (CDC), Maui, HI, 2012, 7485–7491. doi: 10.1109/CDC.2012.6426140. [14] M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. [15] J.-H. Eschenburg and E. Heintze, Extrinsic symmetric spaces and orbits of $s$-representations, Manuscripta Math., 88 (1995), 517-524.  doi: 10.1007/BF02567838. [16] D. Ferus, Immersions with parallel second fundamental form, Math. Z., 140 (1974), 87-93.  doi: 10.1007/BF01218650. [17] M. Godoy Molina, E. Grong, I. Markina and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds, J. Dyn. Control Syst., 18 (2012), 181-214.  doi: 10.1007/s10883-012-9139-2. [18] M. Harandi, R. Hartley, C. Shen, B. Lovell and C. Sanderson, Extrinsic methods for coding and dictionary learning on Grassmann manifolds, Int. J. Comput. Vis., 114 (2015), 113-136.  doi: 10.1007/s11263-015-0833-x. [19] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511811685. [20] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied Mathematics, 80, Academic Press, Inc., New York-London, 1978. [21] S. Helgason, On Riemannian curvature of homogeneous spaces, Proc. Amer. Math. Soc., 9 (1958), 831-838.  doi: 10.1090/S0002-9939-1958-0108811-2. [22] Z. Huang, R. Wang, S. Shan and X. Chen, Projection metric learning on Grassmann manifold with application to video based face recognition, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, 2015,140–149. doi: 10.1109/CVPR.2015.7298609. [23] K. Hüper, K. A. Krakowski and F. Silva Leite, Rolling maps in a Riemannian framework, in Mathematical Papers in Honour of Fátima Silva Leite, Textos Mat. Sér. B, 43, Univ. Coimbra, Coimbra, 2011, 15–30. [24] K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Graßmann manifolds, J. Dyn. Control Syst., 13 (2007), 467-502.  doi: 10.1007/s10883-007-9027-3. [25] V. Jurdjevic and J. Zimmerman, Rolling problems on spaces of constant curvature, in Lagrangian and Hamiltonian Methods for Nonlinear Control, Lect. Notes Control Inf. Sci., 366, Springer, Berlin, 2007,221–231. doi: 10.1007/978-3-540-73890-9_17. [26] V. Jurdjevic and J. Zimmerman, Rolling sphere problems on spaces of constant curvature, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729-747.  doi: 10.1017/S0305004108001084. [27] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 1, Interscience Publishers John Wiley & Sons, Inc.,, New York-London, 1963. [28] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 2, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. [29] A. Korolko and F. Silva Leite, Kinematics for rolling a Lorentzian sphere, 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011. doi: 10.1109/CDC.2011.6160592. [30] O. Kowalski, Generalized Symmetric Spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/BFb0103324. [31] K. A. Krakowski, L. Machado and F. Silva Leite, Rolling symmetric spaces, in Geometric Science of Information, Lecture Notes in Comput. Sci., 9389, Springer, Cham, 2015,550–557. doi: 10.1007/978-3-319-25040-3_59. [32] K. A. Krakowski and F. Silva Leite, An algorithm based on rolling to generate smooth interpolating curves on ellipsoids, Kybernetika, 50 (2014), 544-562.  doi: 10.14736/kyb-2014-4-0544. [33] J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [34] K. Lynch and F. Park, Modern Robotics - Mechanics, Planning, and Control, Cambridge University Press, New York, 2017. [35] L. Machado, F. Pina and F. Silva Leite, Rolling maps for the essential manifold, in Dynamics, Games and Science, CIM Ser. Math. Sci., 1, Springer, Cham, 2015,399–415. doi: 10.1007/978-3-319-16118-1_21. [36] M. A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math, 8 (1984), 31-54.  doi: 10.21099/tkbjm/1496159942. [37] I. Markina and F. Silva Leite, Introduction to the intrinsic rolling with indefinite metric, Comm. Anal. Geom., 24 (2016), 1085-1106.  doi: 10.4310/CAG.2016.v24.n5.a7. [38] A. Marques and F. Silva Leite, Controllability for the constrained rolling motion of symplectic groups, in Proc. of the 11th Portuguese Conference on Automatic Control, Lecture Notes in Electrical Engineering, 321, Springer, Cham, 2015, 3–12. doi: 10.1007/978-3-319-10380-8_1. [39] R. N. Murray, Z. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. [40] K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact, Tohoku Math. J. (2), 30 (1978), 623-637.  doi: 10.2748/tmj/1178229921. [41] B. O'Neill, Elementary Differential Geometry, Academic Press, New York-London, 1966. [42] X. Pennec, S. Sommer and P. T. Fletcher, Riemannian Geometric Statistics in Medical Image Analysis, Academic Press, 2020.  doi: 10.1016/C2017-0-01561-6. [43] A. G. Rojo and A. M. Bloch, The rolling sphere, the quantum spin, and a simple view of the Landau-Zener problem, American J. Physics, 78 (2010), 1014-1022.  doi: 10.1119/1.3456565. [44] R. W. Sharpe, Differential Geometry. Cartan's Generalization of Klein's Erlangen Program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997. [45] M. Takeuchi and S. Kobayashi, Minimal imbeddings of ${R}$-spaces, J. Differential Geometry, 2 (1968), 203-215.  doi: 10.4310/jdg/1214428257. [46] R. Tron and K. Daniilidis, The space of essential matrices as a Riemannian quotient manifold, SIAM J. Imaging Sci., 10 (2017), 1416-1445.  doi: 10.1137/16M1091332. [47] P. Turaga and R. Chellappa, Locally time-invariant models of human activities using trajectories on the Grassmannian, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Miami, FL, 2009, 2435–2441. doi: 10.1109/CVPR.2009.5206710. [48] P. Turaga, A. Veeraraghavan, A. Srivastava and R. Chellappa, Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2273-2286.  doi: 10.1109/TPAMI.2011.52. [49] R. Vemulapalli and R. Chellappa, Rolling rotations for recognizing human actions from 3d skeletal data, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, 2016, 4471–4479. doi: 10.1109/CVPR.2016.484. [50] L. Vrancken, Parallel affine immersions with maximal codimension, Tohoku Math. J. (2), 53, Number 4 (2001), 511–531. [51] J. Zhang, G. Zhu, R. Heath Jr. and K. Huang, Grassmannian learning: Embedding geometry awareness in shallow and deep learning, preprint, arXiv: 1808.02229[cs.LG]. [52] J. A. Zimmerman, Optimal control of the sphere ${S^n}$ rolling on ${E^n}$, Math. Control Signals Systems, 17 (2005), 14-37.  doi: 10.1007/s00498-004-0143-2.

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##### References:
 [1] A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [2] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Springer-Verlag, New York, 2003. doi: 10.1115/1.1641775. [3] A. M. Bloch, R. W. Brockett and P. E. Crouch, Double bracket equations and geodesic flows on symmetric spaces, Comm. Math. Phys., 187 (1997), 357-373.  doi: 10.1007/s002200050140. [4] A. M. Bloch, M. Camarinha and L. Colombo, Variational point-obstacle avoidance on Riemannian manifolds, preprint, arXiv: 1909.12321[eess.SY]. [5] A. M. Bloch and P. E. Crouch, Optimal control, optimization, and analytical mechanics, in Mathematical Control Theory, Springer, New York, 1999,268–321. doi: 10.1007/978-1-4612-1416-8_8. [6] A. M. Bloch, P. E. Crouch and T. S. Ratiu, Sub-Riemannian optimal control problems, in Hamiltonian and Gradient Flows, Algorithms and Control, Fields Inst. Commun., 3, Amer. Math. Soc., Providence, RI, 1994, 35–48. [7] A. M. Bloch and A. G. Rojo, Kinematics of the rolling sphere and quantum spin, Commun. Inf. Syst., 10 (2010), 221-238.  doi: 10.4310/CIS.2010.v10.n4.a4. [8] R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math., 114 (1993), 435-461.  doi: 10.1007/BF01232676. [9] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [10] G. S. Chirikjian, Information theory on Lie groups and mobile robotics applications, IEEE International Conference on Robotics and Automation, Anchorage, AK, 2010, 2751–2757. doi: 10.1109/ROBOT.2010.5509791. [11] Y. Chitour, M. Godoy Molina and P. Kokkonen, The rolling problem: overview and challenges, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser., 5, Springer, Cham, 2014,103–122. doi: 10.1007/978-3-319-02132-4_7. [12] Y. Chitour and P. Kokkonen, Rolling manifolds on space forms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 927-954.  doi: 10.1016/j.anihpc.2012.05.005. [13] P. E. Crouch and F. Silva Leite, Rolling motions of pseudo-orthogonal groups, IEEE 51st Annual Conference on Decision and Control (CDC), Maui, HI, 2012, 7485–7491. doi: 10.1109/CDC.2012.6426140. [14] M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. [15] J.-H. Eschenburg and E. Heintze, Extrinsic symmetric spaces and orbits of $s$-representations, Manuscripta Math., 88 (1995), 517-524.  doi: 10.1007/BF02567838. [16] D. Ferus, Immersions with parallel second fundamental form, Math. Z., 140 (1974), 87-93.  doi: 10.1007/BF01218650. [17] M. Godoy Molina, E. Grong, I. Markina and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds, J. Dyn. Control Syst., 18 (2012), 181-214.  doi: 10.1007/s10883-012-9139-2. [18] M. Harandi, R. Hartley, C. Shen, B. Lovell and C. Sanderson, Extrinsic methods for coding and dictionary learning on Grassmann manifolds, Int. J. Comput. Vis., 114 (2015), 113-136.  doi: 10.1007/s11263-015-0833-x. [19] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511811685. [20] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure and Applied Mathematics, 80, Academic Press, Inc., New York-London, 1978. [21] S. Helgason, On Riemannian curvature of homogeneous spaces, Proc. Amer. Math. Soc., 9 (1958), 831-838.  doi: 10.1090/S0002-9939-1958-0108811-2. [22] Z. Huang, R. Wang, S. Shan and X. Chen, Projection metric learning on Grassmann manifold with application to video based face recognition, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, 2015,140–149. doi: 10.1109/CVPR.2015.7298609. [23] K. Hüper, K. A. Krakowski and F. Silva Leite, Rolling maps in a Riemannian framework, in Mathematical Papers in Honour of Fátima Silva Leite, Textos Mat. Sér. B, 43, Univ. Coimbra, Coimbra, 2011, 15–30. [24] K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Graßmann manifolds, J. Dyn. Control Syst., 13 (2007), 467-502.  doi: 10.1007/s10883-007-9027-3. [25] V. Jurdjevic and J. Zimmerman, Rolling problems on spaces of constant curvature, in Lagrangian and Hamiltonian Methods for Nonlinear Control, Lect. Notes Control Inf. Sci., 366, Springer, Berlin, 2007,221–231. doi: 10.1007/978-3-540-73890-9_17. [26] V. Jurdjevic and J. Zimmerman, Rolling sphere problems on spaces of constant curvature, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729-747.  doi: 10.1017/S0305004108001084. [27] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 1, Interscience Publishers John Wiley & Sons, Inc.,, New York-London, 1963. [28] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics, 2, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. [29] A. Korolko and F. Silva Leite, Kinematics for rolling a Lorentzian sphere, 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011. doi: 10.1109/CDC.2011.6160592. [30] O. Kowalski, Generalized Symmetric Spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/BFb0103324. [31] K. A. Krakowski, L. Machado and F. Silva Leite, Rolling symmetric spaces, in Geometric Science of Information, Lecture Notes in Comput. Sci., 9389, Springer, Cham, 2015,550–557. doi: 10.1007/978-3-319-25040-3_59. [32] K. A. Krakowski and F. Silva Leite, An algorithm based on rolling to generate smooth interpolating curves on ellipsoids, Kybernetika, 50 (2014), 544-562.  doi: 10.14736/kyb-2014-4-0544. [33] J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [34] K. Lynch and F. Park, Modern Robotics - Mechanics, Planning, and Control, Cambridge University Press, New York, 2017. [35] L. Machado, F. Pina and F. Silva Leite, Rolling maps for the essential manifold, in Dynamics, Games and Science, CIM Ser. Math. Sci., 1, Springer, Cham, 2015,399–415. doi: 10.1007/978-3-319-16118-1_21. [36] M. A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math, 8 (1984), 31-54.  doi: 10.21099/tkbjm/1496159942. [37] I. Markina and F. Silva Leite, Introduction to the intrinsic rolling with indefinite metric, Comm. Anal. Geom., 24 (2016), 1085-1106.  doi: 10.4310/CAG.2016.v24.n5.a7. [38] A. Marques and F. Silva Leite, Controllability for the constrained rolling motion of symplectic groups, in Proc. of the 11th Portuguese Conference on Automatic Control, Lecture Notes in Electrical Engineering, 321, Springer, Cham, 2015, 3–12. doi: 10.1007/978-3-319-10380-8_1. [39] R. N. Murray, Z. X. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994. [40] K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact, Tohoku Math. J. (2), 30 (1978), 623-637.  doi: 10.2748/tmj/1178229921. [41] B. O'Neill, Elementary Differential Geometry, Academic Press, New York-London, 1966. [42] X. Pennec, S. Sommer and P. T. Fletcher, Riemannian Geometric Statistics in Medical Image Analysis, Academic Press, 2020.  doi: 10.1016/C2017-0-01561-6. [43] A. G. Rojo and A. M. Bloch, The rolling sphere, the quantum spin, and a simple view of the Landau-Zener problem, American J. Physics, 78 (2010), 1014-1022.  doi: 10.1119/1.3456565. [44] R. W. Sharpe, Differential Geometry. Cartan's Generalization of Klein's Erlangen Program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997. [45] M. Takeuchi and S. Kobayashi, Minimal imbeddings of ${R}$-spaces, J. Differential Geometry, 2 (1968), 203-215.  doi: 10.4310/jdg/1214428257. [46] R. Tron and K. Daniilidis, The space of essential matrices as a Riemannian quotient manifold, SIAM J. Imaging Sci., 10 (2017), 1416-1445.  doi: 10.1137/16M1091332. [47] P. Turaga and R. Chellappa, Locally time-invariant models of human activities using trajectories on the Grassmannian, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Miami, FL, 2009, 2435–2441. doi: 10.1109/CVPR.2009.5206710. [48] P. Turaga, A. Veeraraghavan, A. Srivastava and R. Chellappa, Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2273-2286.  doi: 10.1109/TPAMI.2011.52. [49] R. Vemulapalli and R. Chellappa, Rolling rotations for recognizing human actions from 3d skeletal data, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, 2016, 4471–4479. doi: 10.1109/CVPR.2016.484. [50] L. Vrancken, Parallel affine immersions with maximal codimension, Tohoku Math. J. (2), 53, Number 4 (2001), 511–531. [51] J. Zhang, G. Zhu, R. Heath Jr. and K. Huang, Grassmannian learning: Embedding geometry awareness in shallow and deep learning, preprint, arXiv: 1808.02229[cs.LG]. [52] J. A. Zimmerman, Optimal control of the sphere ${S^n}$ rolling on ${E^n}$, Math. Control Signals Systems, 17 (2005), 14-37.  doi: 10.1007/s00498-004-0143-2.
A sphere $\mathbf{M}$ is rolling upon a surface $\mathbf{M}_0$, along the development curve $\sigma_0$, without slipping or twisting
A vector attached to a rolling manifold generates a vector field along the path $\chi(t)(p)$ in the ambient space; thus a tangent (normal) vector in $\mathbf{T}_p \mathbf{M}$ is isometrically carried to a tangent (normal) vector in $\mathbf{T}_q(\chi(t_0)( \mathbf{M})) = \mathbf{T}_q \mathbf{M}_0$
A sphere ${\bf S}^{2}$ is rolling upon $\mathbf{M}_0$ along the development curve $\sigma_0$ without slipping or twisting; the infinitesimal action $({\dot\chi\,\chi {^{-1}}})_*$ is orthogonal, with respect to an ${\rm Ad}(H)$-invariant inner product onk__ge $\mathfrak{so}({3})$, to the Lie algebra of the isotropy group
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