Geodesics in infinite dimensional Stiefel and Grassmann manifolds

Philipp Harms; Andrea C.G. Mennucci

doi:10.1016/j.crma.2012.08.010

Geometry/Calculus of Variations

Geodesics in infinite dimensional Stiefel and Grassmann manifolds

Presented by: Jean-Pierre Demailly

Philipp Harms ¹; Andrea C.G. Mennucci ²

¹ Harvard Education Innovation Laboratory, Harvard University, 44, Brattle Street, Cambridge, MA 02138, USA
² Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 773-776.

Abstract
Résumé

Let V be a separable Hilbert space, possibly infinite dimensional. Let $St (p, V)$ be the Stiefel manifold of orthonormal frames of p vectors in V, and let $Gr (p, V)$ be the Grassmann manifold of p-dimensional subspaces of V. We study the distance and the geodesics in these manifolds, by reducing the matter to the finite dimensional case. We then prove that any two points in those manifolds can be connected by a minimal geodesic, and characterize the cut locus.

Soit V un espace de Hilbert séparable, éventuellement de dimension infinie. Soient $St (p, V)$ lʼensemble des systèmes orthonormés de p vecteurs de V, appelé la variété de Stiefel, et $Gr (p, V)$ lʼensemble des sous-espaces vectoriels de V de dimension p, appelé la variété Grassmannienne. En réduisant le problème en dimension finie, nous montrons que dans ces espaces il existe des géodésiques minimales entre chaque paire de points et nous caractérisons le cut-locus.

Received: 2010-11-15
Accepted: 2012-08-30
Published online: 2012-08-01

DOI: 10.1016/j.crma.2012.08.010

Author's affiliations:

Philipp Harms ¹; Andrea C.G. Mennucci ²

¹ Harvard Education Innovation Laboratory, Harvard University, 44, Brattle Street, Cambridge, MA 02138, USA
² Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

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     author = {Philipp Harms and Andrea C.G. Mennucci},
     title = {Geodesics in infinite dimensional {Stiefel} and {Grassmann} manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {773--776},
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     year = {2012},
     doi = {10.1016/j.crma.2012.08.010},
     language = {en},
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Philipp Harms; Andrea C.G. Mennucci. Geodesics in infinite dimensional Stiefel and Grassmann manifolds. Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 773-776. doi : 10.1016/j.crma.2012.08.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.010/

References
Cited by

[1] C.J. Atkin The Hopf–Rinow theorem is false in infinite dimensions, Bull. London Math. Soc., Volume 7 (1975) no. 3, pp. 261-266 | DOI

[2] A. Edelman; T. Arias; S. Smith The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., Volume 20 (1998), pp. 303-353 | arXiv | DOI

[3] N. Grossman Hilbert manifolds without epiconjugate points, Proc. Amer. Math. Soc., Volume 16 (1965), pp. 1365-1371 (ISSN: 0002-9939) | DOI

[4] S. Lang Fundamentals of Differential Geometry, Springer-Verlag, 1999 (ISBN: 0-387-98593-X)

[5] G. Sundaramoorthi, A. Mennucci, S. Soatto, A. Yezzi, Tracking deforming objects by filtering and prediction in the space of curves, in: Conference on Decision and Control, ISBN 978-1-4244-3871-6, 2009, pp. 2395–2401, . | DOI

[6] G. Sundaramoorthi; A. Mennucci; S. Soatto; A. Yezzi A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering, SIAM J. Imaging Sci., Volume 4 (2011), pp. 109-145 | DOI

[7] L. Younes Computable elastic distances between shapes, SIAM J. Appl. Math., Volume 58 (1998) no. 2, pp. 565-586 | DOI

[8] L. Younes; P.W. Michor; J. Shah; D. Mumford A metric on shape space with explicit geodesics, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., Volume 19 (2008) no. 1, pp. 25-57 (ISSN: 1120-6330) | DOI

Cited by Sources:

^☆ This research was funded by SNS09MENNB of the Scuola Normale Superiore, and by the FWF Project 21030.

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