Sobolev metrics on the manifold of all riemannian metrics

Journal of Differential Geometry

Volumn 94, Issue 2, 2013, Pages 187-208

  Bauer, Martin ^a   Harms, Philipp ^b   Michor, Peter W ^a  

^a UNIVERSITY OF VIENNA   (Austria)

^b HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84879998697     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1367438647     Document Type: Article

References (30)

1. M. Bauer & M. Bruveris, A new Riemannian setting for surface registration, in 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, pages 182-194, 2011.
2. M. Bauer, P. Harms & P.W. Michor, Almost local metrics on shape space of hypersurfaces in n-space, SIAM J. Imaging Sci. 5 (2012), pp. 244-310.
3. M. Bauer, P. Harms & P.W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics 3, 4 (2011).
4. M. Bauer, P. Harms & P.W. Michor, Curvature weighted metrics on shape space of hypersurfaces in n-space, Differential Geometry and Its Applications, 30 (2012), 33-41.
5. M. Bauer, M. Bruveris, P. Harms & P.W. Michor, The KdV-equation has vanishing geodesic distance, Ann. Glob. Anal. Geom. 41, 4 (2012) 461-472.
6. A.L. Besse Einstein manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008.
7. J. Cheeger, M. Gromov & M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17, 15-53, 1982.
8. 2 Metric, Doctoral Dissertation, Leipzig 2009.
9. B. Clarke, The Metric Geometry of the Manifold of Riemannian Metrics over a Closed Manifold, Calc. Var. Partial Differential Equations, 39, 533-545, 2010.
10. B. Clarke, The Completion of the Manifold of Riemannian Metrics, J. Differential Geom., 93 (2), 203-268, 2013.
11. D.G. Ebin, The manifold of Riemannian metrics, Global Analysis, Proc. Symp. Pure Math. AMS, 15, 11-40, 1970.
12. J. Eichhorn, Global analysis on open manifolds, Nova Science Publishers, Inc., New York, 2007.
13. J. Eichhorn & J. Fricke, The module structure theorem for Sobolev spaces on open manifolds, Math. Nachr., 194, 35-47, 1998.
14. D.S. Freed & D. Groisser, The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group, Michigan Math. J., 36, 323-344, 1989.
15. O. Gil-Medrano & P.W. Michor, The Riemannian manifold of all Riemannian metrics, Quarterly J. Math. Oxford, 42, 183-202, 1991.
16. O. Gil-Medrano, P.W. Michor & M. Neuwirther, Pseudoriemannian metrics on spaces of bilinear structures, Quarterly J. Math. Oxford, 43, 201-221, 1992.
17. G. Gil-Medrano & P.W. Michor, Geodesics on spaces of almost hermitian structures, Israel J. Math., 88, 319-332, 1994.
18. S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol.1, Wiley Classics Library. John Wiley & Sons Inc., New York, 1996.
19. I. Kolar, P.W. Michor & J. Slovak, Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
20. A. Kriegl & P.W. Michor, The convenient setting of global analysis, volume 53 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.
21. A.C.G. Mennucci, A. Yezzi & G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound., 10 (4), 423-445, 2008.
22. P.W. Michor. Topics in differential geometry, volume 93 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2008.
23. P.W. Michor & D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math., 10, 217-245 (electronic), 2005.
24. P.W. Michor & D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS) 8, (2006), 1-48, 2006.
25. P.W. Michor & D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (1), 74-113, 2007.
26. M.A. Shubin, Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987.
27. A. Trouve & L. Younes, Diffeomorphic matching problems in one dimension: Designing and minimizing matching functionals, in David Vernon, editor, Computer Vision, volume 1842. ECCV, 2000.
28. S. Verpoort, The geometry of the second fundamental form: Curvature properties and variational aspects, PhD thesis, Katholieke Universiteit Leuven, 2008.
29. L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (2): 565-586 (electronic), 1998.
30. 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., 19 (1): 25-57, 2008.