Proceedings of the Japan Academy, Series A, Mathematical Sciences

Riemannian submersions, minimal immersions and cohomology class

Bang-Yen Chen

Full-text: Open access

Abstract

We prove a simple optimal relationship between Riemannian submersions and minimal immersions; namely, if a Riemannian manifold admits a non-trivial Riemannian submersion with totally geodesic fibers, then it cannot be isometrically immersed in any Riemannian manifold of non-positive sectional curvature as a minimal manifold. Some related results are also presented. In the last section, we introduce a cohomology class for Riemannian submersions and provide an application.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 10 (2005), 162-167.

Dates
First available in Project Euclid: 28 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1135791768

Digital Object Identifier
doi:10.3792/pjaa.81.162

Mathematical Reviews number (MathSciNet)
MR2196721

Zentralblatt MATH identifier
1147.53312

Subjects
Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]

Keywords
Riemannian submersion minimal immersion cohomology class totally geodesic fibers

Citation

Chen, Bang-Yen. Riemannian submersions, minimal immersions and cohomology class. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 162--167. doi:10.3792/pjaa.81.162. https://projecteuclid.org/euclid.pja/1135791768


Export citation

References

  • S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, Princeton Univ. Press, Princeton, NJ, 1982.
  • L. Amour, R. Brummelhuis and J. Nourrigat, Resonances of the Dirac Hamiltonian in the non relativistic limit, Ann. Henri Poincaré 2 (2001), no. 3, 583–603.
  • R. J. Cirincione and P. R. Chernoff, Dirac and Klein-Gordon equations: convergence of solutions in the nonrelativistic limit, Comm. Math. Phys. 79 (1981), no. 1, 33–46.
  • D. R. Grigore, G. Nenciu and R. Purice, On the nonrelativistic limit of the Dirac Hamiltonian, Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), no. 3, 231–263.
  • B. Helffer and J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper, in Schrödinger operators (Sønderborg, 1988), 118–197, Lecture Notes in Phys., 345, Springer, Berlin, (1989).
  • W. Hunziker, On the nonrelativistic limit of the Dirac theory, Comm. Math. Phys. 40 (1975), 215–222.
  • T. Ikebe and T. Kato, Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Rational Mech. Anal. 9 (1962), 77–92.
  • Isozaki, H., Many-body Schrödinger equations, Springer-Verlag, Tokyo, 2004. (In Japanese).
  • H. Kalf, T. Ōkaji and O. Yamada, Absence of eigenvalues of Dirac operators with potentials diverging at infinity, Math. Nachr. 259 (2003), 19–41.
  • M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York, 1972.
  • M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York, 1978.
  • K. M. Schmidt and O. Yamada, Spherically symmetric Dirac operators with variable mass and potentials infinite at infinity, Publ. Res. Inst. Math. Sci. 34 (1998), no. 3, 211–227.
  • Z. Shen, Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4465–4488.
  • E. C. Titchmarsh, A problem in relativistic quantum mechanics, Proc. London Math. Soc. (3) 11 (1961), 169–192.
  • J. Uchiyama and O. Yamada, Sharp estimates of lower bounds of polynomial decay order of eigenfunctions, Publ. Res. Inst. Math. Sci. 26 (1990), no. 3, 419–449.
  • K. Veselić, The nonrelativistic limit of the Dirac equation and the spectral concentration, Glasnik Mat. Ser. III 4 (24) (1969), 231–241.
  • K. Yajima, Nonrelativistic limit of the Dirac theory, scattering theory, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 517–523.
  • O. Yamada, On the spectrum of Dirac operators with the unbounded potential at infinity, Hokkaido Math. J. 26 (1997), no. 2, 439–449. \beginthebibliography99
  • B.-Y. Chen, Geometry of submanifolds, Dekker, New York, 1973.
  • B.-Y. Chen, Geometry of submanifolds and its applications, Sci. Univ. Tokyo, Tokyo, 1981.
  • B.-Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel) 60 (1993), no. 6, 568–578.
  • B.-Y. Chen, Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. (N.S.) 26 (2000), no. 1, 105–127.
  • B.-Y. Chen, Examples and classification of Riemannian submersions satisfying a basic equality. Bull. Austral. Math. Soc. 72 (2005), 391–402.
  • J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
  • N. Ejiri, Minimal immersions of Riemannian products into real space forms, Tokyo J. Math. 2 (1979), no. 1, 63–70.
  • P. B. Gilkey, J. V. Leahy and J. Park, Spinors, spectral geometry, and Riemannian submersions, Seoul Nat. Univ., Seoul, 1998.
  • J. D. Moore, Isometric immersions of riemannian products, J. Differential Geometry 5 (1971), 159–168.
  • T. Nagano, On fibred Riemann manifolds, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 10 (1960), 17–27.
  • B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469.
  • K. Yano and S. Ishihara, Tangent and cotangent bundles: differential geometry, Dekker, New York, 1973.