Abstract
In this work we study the kinematics of an m-dimensional continuum moving in an (m+1)-dimensional ambient space, modelled by the motion of a hypersurface M in a Riemannian manifold N. Focusing on the codimension of the continuum relative to the ambient space rather than on its dimension, we provide a unified framework for either curves moving on surfaces, surfaces moving in a 3-dimensional space (Euclidean or Riemannian), or hypersurfaces moving in a Riemannian space of arbitrary dimension. Further, the use of general geometric structures and a coordinate-free language facilitate the description of more general continua. We present formulae for the variation of geometric quantities of the hypersurface, some of which generalize those given for surfaces in Euclidean space (Kadianakis, N. in J. Elasticity 16:1–17, 2010). The main point in the present work is the use of kinematical quantities which result from a generalized version of the polar decomposition theorem for surfaces introduced by Man, C.-S. and Cohen, H. (in J. Elast. 16:97–104, 1986) and adapted here for hypersurfaces. We apply our results to motion along the normal and to pure strain motion. Finally, we discuss the relation of our results to Differential Geometry and Physics. Specifically, we provide an application for the case of a curve moving on a 2-dimensional manifold, which models a 1-dimensional continuum moving on a surface. The later application is presented independently of the embedding of the surface in a larger space.
Similar content being viewed by others
References
Kadianakis, N.: Evolution of surfaces and the kinematics of membranes. J. Elast. 99, 1–17 (2010)
Man, C.-S., Cohen, H.: A coordinate-free approach to the kinematics of membranes. J. Elast. 16, 97–104 (1986)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Murdoch, A.I.: A coordinate-free approach to surface kinematics. Glasg. Math. J. 32, 299–307 (1990)
Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2, 197–226 (1958)
Truesdell, C.: A First Course in Rational Continuum Mechanics, vol. 1. Academic Press, San Diego (1977)
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983)
Segev, R., Rodnay, G.: Cauchy’s theorem on manifolds. J. Elast. 56, 129–144 (1999)
Fosdick, R., Tang, H.: Surface transport in continuum mechanics. Math. Mech. Solids 14, 587–598 (2009)
Kadianakis, N.: On the geometry of Lagrangian and Eulerian descriptions in continuum mechanics. Z. Angew. Math. Mech. 79, 131–138 (1999)
Appleby, P.G., Kadianakis, N.: A frame-independent description of the principles of classical mechanics. Arch. Ration. Mech. Anal. 95, 1–22 (1986)
Capriz, G.: Continua with Microstructure. Springer, Berlin (1989)
Epstein, M., Manuel de Leon, B.: Geometrical theory of uniform Cosserat media. J. Geom. Phys. 26, 127–170 (1998)
Yavari, A., Marsden, J.E.: Covariant balance laws in continua with microstructure. Rep. Math. Phys. 63(1), 1–42 (2009)
Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Differ. Geom. 39, 407–431 (1994)
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)
Capovilla, R., Guven, J., Santiago, J.A.: Deformations of the geometry of lipid vesicles. J. Phys. A, Math. Gen. 36, 6281–6295 (2003)
Capovilla, R., Guven, J.: Geometry of deformations of relativistic membranes. Phys. Rev. D 51(12), 6736–6743 (1995)
Goldstein, R.A., Ryan, P.J.: Infinitesimal rigidity of Euclidean submanifolds. J. Differ. Geom. 10, 49–60 (1975)
Yano, K.: Integral Formulas in Riemannian Geometry. Dekker, New York (1970)
do Carmo, M.: Riemannian Geometry. Birkhauser, Boston (1992)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 4. Publish or Perish, Boston (1979)
Szwabowicz, M.L.: Pure strain deformations of surfaces. J. Elast. 92, 255–275 (2008)
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)
Langer, R., Singer, D.: The total squared curvature of closed curves. J. Differ. Geom. 20(1), 1–22 (1984)
Hangan, T., Murea, C.M., Sari, T.: Poleni curves on surfaces of constant curvature. Rend. Semin. Mat. (Torino) 67(1), 91–107 (2009)
Arroyo, J., Garay, O.J., Mencia, J.J.: Closed generalized elastic curves in S 2(1). J. Geom. Phys. 48, 339–353 (2003)
Maekawa, T.: An overview of offset curves and surfaces. Comput. Aided Geom. Des. 31, 165–173 (1999)
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)
John, F.: Partial Differential Equations, 3rd edn. Springer, Berlin (1971)
Barbosa, J.L., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in riemannian manifolds. Math. Z. 197, 123–138 (1988)
Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984)
Acknowledgements
The authors are grateful to Daniel Meyer of the Institut de Mathematiques de Jussieu, Universite Paris 7, for his helpful discussions and critical comments during the preparation of this work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kadianakis, N., Travlopanos, F. Kinematics of Hypersurfaces in Riemannian Manifolds. J Elast 111, 223–243 (2013). https://doi.org/10.1007/s10659-012-9399-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-012-9399-9