# American Institute of Mathematical Sciences

September  2016, 8(3): 323-357. doi: 10.3934/jgm.2016010

## Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

 1 Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany 2 Head of Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany

Received  October 2014 Revised  February 2016 Published  September 2016

We provide an easy approach to the geodesic distance on the general linear group $GL(n)$ for left-invariant Riemannian metrics which are also right-$O(n)$-invariant. The parameterization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated.
Citation: Robert J. Martin, Patrizio Neff. Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics. Journal of Geometric Mechanics, 2016, 8 (3) : 323-357. doi: 10.3934/jgm.2016010
##### References:
 [1] E. Andruchow, G. Larotonda, L. Recht and A. Varela, The left invariant metric in the general linear group,, Journal of Geometry and Physics, 86 (2014), 241. doi: 10.1016/j.geomphys.2014.08.009. Google Scholar [2] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition),, Princeton reference, (2009). doi: 10.1515/9781400833344. Google Scholar [3] A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups,, Journal of Geometric Mechanics, 3 (2011), 197. doi: 10.3934/jgm.2011.3.197. Google Scholar [4] P. G. Ciarlet, Three-Dimensional Elasticity,, Number 1 in Studies in Mathematics and its Applications, (1988). Google Scholar [5] C. De Boor, A naive proof of the representation theorem for isotropic, linear asymmetric stress-strain relations,, Journal of Elasticity, 15 (1985), 225. doi: 10.1007/BF00041995. Google Scholar [6] M. P. do Carmo, Riemannian Geometry,, Birkhäuser Basel, (1992). doi: 10.1007/978-1-4757-2201-7. Google Scholar [7] J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen,, Springer, (2007). Google Scholar [8] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (1990). doi: 10.1007/978-3-642-97242-3. Google Scholar [9] H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?,, Zeitschrift für Physik, 55 (1929), 145. Google Scholar [10] N. J. Higham, Functions of Matrices: Theory and Computation,, Society for Industrial and Applied Mathematics, (2008). doi: 10.1137/1.9780898717778. Google Scholar [11] J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.),, Springer, (1998). doi: 10.1007/978-3-662-22385-7. Google Scholar [12] K. Königsberger, Analysis 1,, Analysis, (2004). Google Scholar [13] J. Lankeit, P. Neff and Y. Nakatsukasa, The minimization of matrix logarithms: On a fundamental property of the unitary polar factor,, Linear Algebra and its Applications, 449 (2014), 28. doi: 10.1016/j.laa.2014.02.012. Google Scholar [14] S. Lee, M. Choi, H. Kim and F. C. Park, Geometric direct search algorithms for image registration,, IEEE Transactions on Image Processing, 16 (2007), 2215. doi: 10.1109/TIP.2007.901809. Google Scholar [15] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17., Springer, (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [16] R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$,, in preparation, (2016). Google Scholar [17] A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$,, in Geometry, (2002), 61. doi: 10.1007/0-387-21791-6_2. Google Scholar [18] M. Moakher, Means and averaging in the group of rotations,, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 1. doi: 10.1137/S0895479801383877. Google Scholar [19] P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations,, Technical report, (2008). Google Scholar [20] P. Neff, B. Eidel and R. J. Martin, Geometry of logarithmic strain measures in solid mechanics,, Archive for Rational Mechanics and Analysis, 222 (2016), 507. doi: 10.1007/s00205-016-1007-x. Google Scholar [21] P. Neff, B. Eidel, F. Osterbrink and R. Martin, A Riemannian approach to strain measures in nonlinear elasticity,, Comptes Rendus Mécanique, 342 (2014), 254. doi: 10.1016/j.crme.2013.12.005. Google Scholar [22] P. Neff, Y. Nakatsukasa and A. Fischle, A logarithmic minimization property of the unitary polar factor in the spectral and frobenius norms,, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1132. doi: 10.1137/130909949. Google Scholar [23] C. H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature,, Oxford Graduate Texts in Mathematics, (2011). doi: 10.1093/acprof:oso/9780199605880.001.0001. Google Scholar [24] B. Vandereycken, P.-A. Absil and S. Vandewalle, A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank,, IMA Journal of Numerical Analysis, 33 (2013), 481. doi: 10.1093/imanum/drs006. Google Scholar

show all references

##### References:
 [1] E. Andruchow, G. Larotonda, L. Recht and A. Varela, The left invariant metric in the general linear group,, Journal of Geometry and Physics, 86 (2014), 241. doi: 10.1016/j.geomphys.2014.08.009. Google Scholar [2] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition),, Princeton reference, (2009). doi: 10.1515/9781400833344. Google Scholar [3] A. M. Bloch, P. E. Crouch, N. Nordkvist and A. K. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups,, Journal of Geometric Mechanics, 3 (2011), 197. doi: 10.3934/jgm.2011.3.197. Google Scholar [4] P. G. Ciarlet, Three-Dimensional Elasticity,, Number 1 in Studies in Mathematics and its Applications, (1988). Google Scholar [5] C. De Boor, A naive proof of the representation theorem for isotropic, linear asymmetric stress-strain relations,, Journal of Elasticity, 15 (1985), 225. doi: 10.1007/BF00041995. Google Scholar [6] M. P. do Carmo, Riemannian Geometry,, Birkhäuser Basel, (1992). doi: 10.1007/978-1-4757-2201-7. Google Scholar [7] J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen,, Springer, (2007). Google Scholar [8] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (1990). doi: 10.1007/978-3-642-97242-3. Google Scholar [9] H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?,, Zeitschrift für Physik, 55 (1929), 145. Google Scholar [10] N. J. Higham, Functions of Matrices: Theory and Computation,, Society for Industrial and Applied Mathematics, (2008). doi: 10.1137/1.9780898717778. Google Scholar [11] J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.),, Springer, (1998). doi: 10.1007/978-3-662-22385-7. Google Scholar [12] K. Königsberger, Analysis 1,, Analysis, (2004). Google Scholar [13] J. Lankeit, P. Neff and Y. Nakatsukasa, The minimization of matrix logarithms: On a fundamental property of the unitary polar factor,, Linear Algebra and its Applications, 449 (2014), 28. doi: 10.1016/j.laa.2014.02.012. Google Scholar [14] S. Lee, M. Choi, H. Kim and F. C. Park, Geometric direct search algorithms for image registration,, IEEE Transactions on Image Processing, 16 (2007), 2215. doi: 10.1109/TIP.2007.901809. Google Scholar [15] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, volume 17., Springer, (1999). doi: 10.1007/978-0-387-21792-5. Google Scholar [16] R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$,, in preparation, (2016). Google Scholar [17] A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$,, in Geometry, (2002), 61. doi: 10.1007/0-387-21791-6_2. Google Scholar [18] M. Moakher, Means and averaging in the group of rotations,, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 1. doi: 10.1137/S0895479801383877. Google Scholar [19] P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations,, Technical report, (2008). Google Scholar [20] P. Neff, B. Eidel and R. J. Martin, Geometry of logarithmic strain measures in solid mechanics,, Archive for Rational Mechanics and Analysis, 222 (2016), 507. doi: 10.1007/s00205-016-1007-x. Google Scholar [21] P. Neff, B. Eidel, F. Osterbrink and R. Martin, A Riemannian approach to strain measures in nonlinear elasticity,, Comptes Rendus Mécanique, 342 (2014), 254. doi: 10.1016/j.crme.2013.12.005. Google Scholar [22] P. Neff, Y. Nakatsukasa and A. Fischle, A logarithmic minimization property of the unitary polar factor in the spectral and frobenius norms,, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1132. doi: 10.1137/130909949. Google Scholar [23] C. H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature,, Oxford Graduate Texts in Mathematics, (2011). doi: 10.1093/acprof:oso/9780199605880.001.0001. Google Scholar [24] B. Vandereycken, P.-A. Absil and S. Vandewalle, A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank,, IMA Journal of Numerical Analysis, 33 (2013), 481. doi: 10.1093/imanum/drs006. Google Scholar
 [1] Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015 [2] Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104. [3] Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365 [4] Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028 [5] John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 [6] David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477 [7] Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 [8] François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39 [9] Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049 [10] Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684 [11] Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275 [12] Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037 [13] Kamil Otal, Ferruh Özbudak. Explicit constructions of some non-Gabidulin linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 589-600. doi: 10.3934/amc.2016028 [14] Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 [15] Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841 [16] Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157 [17] Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99 [18] Randall Dougherty and Thomas Jech. Left-distributive embedding algebras. Electronic Research Announcements, 1997, 3: 28-37. [19] Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261 [20] Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster. A symmetric nearly preserving general linear method for Hamiltonian problems. Conference Publications, 2015, 2015 (special) : 330-339. doi: 10.3934/proc.2015.0330

2018 Impact Factor: 0.525