Kinematics of Hypersurfaces in Riemannian Manifolds
Source abstract
Abstract
Relying on the generalized version of the polar decomposition theorem for surfaces proven by \textit{C.-S. Man} and \textit{H. Cohen} [J. Elasticity 16, 97--104 (1986; Zbl 0582.73011)], the authors make a deep study of the kinematics of an \(m\)-dimensional continuum moving in an ambient space of one more dimension. As special cases they consider the motion of a curve on a two-dimensional manifold and the problem of Euler elastica. The paper ends with some comment on the geometry and physics of the constant mean curvature hypersurfaces.
Source: openaire · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: Geometric Analysis and Curvature Flows · Advanced Differential Geometry Research · Elasticity and Material Modeling
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (32)
Show source-provided references
1. Kadianakis, N.: Evolution of surfaces and the kinematics of membranes. J. Elast. 99, 1–17 (2010)10.1007/s10659-009-9226-0 ↗
2. Man, C.-S., Cohen, H.: A coordinate-free approach to the kinematics of membranes. J. Elast. 16, 97–104 (1986)10.1007/BF00041068 ↗
3. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)10.1007/BF00261375 ↗
4. Murdoch, A.I.: A coordinate-free approach to surface kinematics. Glasg. Math. J. 32, 299–307 (1990)10.1017/S0017089500009381 ↗
5. Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2, 197–226 (1958)10.1007/BF00277929 ↗
6. Truesdell, C.: A First Course in Rational Continuum Mechanics, vol. 1. Academic Press, San Diego (1977)
7. Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983)
8. Segev, R., Rodnay, G.: Cauchy’s theorem on manifolds. J. Elast. 56, 129–144 (1999)10.1023/A:1007651917362 ↗
9. Fosdick, R., Tang, H.: Surface transport in continuum mechanics. Math. Mech. Solids 14, 587–598 (2009)10.1177/1081286507087316 ↗
10. Kadianakis, N.: On the geometry of Lagrangian and Eulerian descriptions in continuum mechanics. Z. Angew. Math. Mech. 79, 131–138 (1999)10.1002/(SICI)1521-4001(199902)79:2<131::AID-ZAMM131>3.0.CO;2-Q ↗
11. Appleby, P.G., Kadianakis, N.: A frame-independent description of the principles of classical mechanics. Arch. Ration. Mech. Anal. 95, 1–22 (1986)10.1007/BF00280786 ↗
12. Capriz, G.: Continua with Microstructure. Springer, Berlin (1989)10.1007/978-1-4612-3584-2 ↗
13. Epstein, M., Manuel de Leon, B.: Geometrical theory of uniform Cosserat media. J. Geom. Phys. 26, 127–170 (1998)10.1016/S0393-0440(97)00042-9 ↗
14. Yavari, A., Marsden, J.E.: Covariant balance laws in continua with microstructure. Rep. Math. Phys. 63(1), 1–42 (2009)10.1016/S0034-4877(09)00007-X ↗
15. Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Differ. Geom. 39, 407–431 (1994)10.4310/jdg/1214454878 ↗
16. Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. In Hildebrandt, S., Struve, M. (Eds.), Calculus of Variations and Geometric Evolution Problems, pp. 45–84. Springer, Berlin (1999)
17. Capovilla, R., Guven, J., Santiago, J.A.: Deformations of the geometry of lipid vesicles. J. Phys. A, Math. Gen. 36, 6281–6295 (2003)10.1088/0305-4470/36/23/301 ↗
18. Capovilla, R., Guven, J.: Geometry of deformations of relativistic membranes. Phys. Rev. D 51(12), 6736–6743 (1995)10.1103/PhysRevD.51.6736 ↗
19. Goldstein, R.A., Ryan, P.J.: Infinitesimal rigidity of Euclidean submanifolds. J. Differ. Geom. 10, 49–60 (1975)10.4310/jdg/1214432675 ↗
20. Yano, K.: Integral Formulas in Riemannian Geometry. Dekker, New York (1970)
21. do Carmo, M.: Riemannian Geometry. Birkhauser, Boston (1992)10.1007/978-1-4757-2201-7 ↗
22. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 4. Publish or Perish, Boston (1979)
23. Szwabowicz, M.L.: Pure strain deformations of surfaces. J. Elast. 92, 255–275 (2008)10.1007/s10659-008-9161-5 ↗
24. van der Heijden, G.H.M.: The static deformation of a twisted elastic rod constrained to lie on a cylinder. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 457, 695–715 (2001)10.1098/rspa.2000.0688 ↗
25. Langer, R., Singer, D.: The total squared curvature of closed curves. J. Differ. Geom. 20(1), 1–22 (1984)10.4310/jdg/1214438990 ↗
26. Hangan, T., Murea, C.M., Sari, T.: Poleni curves on surfaces of constant curvature. Rend. Semin. Mat. (Torino) 67(1), 91–107 (2009)
27. Arroyo, J., Garay, O.J., Mencia, J.J.: Closed generalized elastic curves in S 2(1). J. Geom. Phys. 48, 339–353 (2003)10.1016/S0393-0440(03)00047-0 ↗
28. Maekawa, T.: An overview of offset curves and surfaces. Comput. Aided Geom. Des. 31, 165–173 (1999)10.1016/S0010-4485(99)00013-5 ↗
29. Simo, J.C., Marsden, J.E.: On the rotated stress tensor and the material version of the Doyle-Ericksen formula. Arch. Ration. Mech. Anal. 86, 213–231 (1984)10.1007/BF00281556 ↗
30. John, F.: Partial Differential Equations, 3rd edn. Springer, Berlin (1971)10.1007/978-1-4615-9966-1 ↗
31. Barbosa, J.L., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in riemannian manifolds. Math. Z. 197, 123–138 (1988)10.1007/BF01161634 ↗
32. Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984)10.1007/BF01215045 ↗