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

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

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

Robert J. Martin ^1, and Patrizio Neff ^2,

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

Abstract
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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.

Keywords: left-invariance, isotropy., general linear group, geodesic distance, Geodesics.

Mathematics Subject Classification: 53C2.

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.
[2]	D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition),, Princeton reference, (2009). doi: 10.1515/9781400833344.
[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.
[4]	P. G. Ciarlet, Three-Dimensional Elasticity,, Number 1 in Studies in Mathematics and its Applications, (1988).
[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.
[6]	M. P. do Carmo, Riemannian Geometry,, Birkhäuser Basel, (1992). doi: 10.1007/978-1-4757-2201-7.
[7]	J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen,, Springer, (2007).
[8]	S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (1990). doi: 10.1007/978-3-642-97242-3.
[9]	H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?,, Zeitschrift für Physik, 55 (1929), 145.
[10]	N. J. Higham, Functions of Matrices: Theory and Computation,, Society for Industrial and Applied Mathematics, (2008). doi: 10.1137/1.9780898717778.
[11]	J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.),, Springer, (1998). doi: 10.1007/978-3-662-22385-7.
[12]	K. Königsberger, Analysis 1,, Analysis, (2004).
[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.
[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.
[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.
[16]	R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$,, in preparation, (2016).
[17]	A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$,, in Geometry, (2002), 61. doi: 10.1007/0-387-21791-6_2.
[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.
[19]	P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations,, Technical report, (2008).
[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.
[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.
[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.
[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.
[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.

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.
[2]	D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas (Second Edition),, Princeton reference, (2009). doi: 10.1515/9781400833344.
[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.
[4]	P. G. Ciarlet, Three-Dimensional Elasticity,, Number 1 in Studies in Mathematics and its Applications, (1988).
[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.
[6]	M. P. do Carmo, Riemannian Geometry,, Birkhäuser Basel, (1992). doi: 10.1007/978-1-4757-2201-7.
[7]	J.-H. Eschenburg and J. Jost, Differentialgeometrie und Minimalflächen,, Springer, (2007).
[8]	S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (1990). doi: 10.1007/978-3-642-97242-3.
[9]	H. Hencky, Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?,, Zeitschrift für Physik, 55 (1929), 145.
[10]	N. J. Higham, Functions of Matrices: Theory and Computation,, Society for Industrial and Applied Mathematics, (2008). doi: 10.1137/1.9780898717778.
[11]	J. Jost, Riemannian Geometry and Geometric Analysis (2nd ed.),, Springer, (1998). doi: 10.1007/978-3-662-22385-7.
[12]	K. Königsberger, Analysis 1,, Analysis, (2004).
[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.
[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.
[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.
[16]	R. J. Martin and P. Neff, The GL(n)-geodesic distance on $SO(n)$,, in preparation, (2016).
[17]	A. Mielke, Finite elastoplasticity, Lie groups and geodesics on $SL(d)$,, in Geometry, (2002), 61. doi: 10.1007/0-387-21791-6_2.
[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.
[19]	P. Neff, Convexity and coercivity in nonlinear, anisotropic elasticity and some useful relations,, Technical report, (2008).
[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.
[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.
[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.
[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.
[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.

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