Skip to main content
SpringerLink
Log in

Spectral zeta function on pseudo H-type nilmanifolds

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), Suppl. I, 3–38.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Agrachev, U. Boscain, J-P. Gauthier and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal., 265 (2009), 2621–2655.

    Article  MathSciNet  Google Scholar 

  3. W. Bauer and K. Furutani, Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zeta-regularized determinant, J. Geom. Phys., 60 (2009), 1209–1234.

    Article  MathSciNet  Google Scholar 

  4. W. Bauer and K. Furutani, Spectral analysis and geometry of a sub-Riemannian structure on S3 and S7, J. Geom. Phys., 58 (2008), 1693–1738.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Bauer, K. Furutani and C. Iwasaki, Fundamental solution for a higher step Grushin type operator, Adv. Math., 271 (2015), 188–234.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Bauer, K. Furutani and C. Iwasaki, Trivializable sub-Riemannian structures on spheres, Bull. Sci. Math., 137(3) (2013), 361–385.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Bauer, K. Furutani and C. Iwasaki, Spectral zeta function of the sub-Laplacian on two step nilmanifolds, J. Math. Pures Appl. (9), 97(3) (2012), 242–261.

    Article  MATH  MathSciNet  Google Scholar 

  8. W. Bauer, K. Furutani and C. Iwasaki, Spectral analysis and geometry of a sub-Laplacian and related Grushin type operators, in: “Partial Differential Equations and Spectral Theory”, Operator Theory: Advances and Applications, 211, 183–287, Springer 2011.

    MathSciNet  Google Scholar 

  9. W. Bauer, K. Furutani and C. Iwasaki, Isospectral, but non-diffeomorphic nilmanifolds with respect to sub-Laplacians and Laplacians, in preparation.

  10. R. Beals, B. Gaveau and P. Greiner, The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemannian complexes, Adv. in Math., 121 (1996), 288–345.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Beals, B. Gaveau and P. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl., 79(7) (2000), 633–689.

    Article  MATH  MathSciNet  Google Scholar 

  12. W. L. Chow, Ü ber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117(1) (1939), 98–105.

    MathSciNet  Google Scholar 

  13. P. Ciatti, Scalar products on Clifford modules and pseudo-H-type Lie algebras, Ann. Mat. Pura Appl., 178(4) (2000), 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Crandall and J. Dodziuk, Integral structures on H-type Lie algebras, J. Lie Theory, 12(1) (2002), 69–79.

    MATH  MathSciNet  Google Scholar 

  15. K. Furutani, Heat kernels of the sub-Laplacian and Laplacian on nilpotent Lie groups, Analysis, Geometry and Topology of Elliptic Operators, papers in honor of Krzysztof P. Wojciechowsky, (World Scientific, London-Singapore) (2006), 185–226.

    MATH  Google Scholar 

  16. K. Furutani and I. Markina, Existence of lattices on general H-type groups, J. Lie Theory, 24 (2014), 979–1011.

    MATH  MathSciNet  Google Scholar 

  17. K. Furutani, C. Iwasaki and T. Kagawa, An action function for a higher step Grushin operator, J. Geom. Phys., 62 (2012), 1949–1976.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Godoy Molina and I. Markina, Sub-Riemannian geodesics and heat operator on odd dimensional spheres, Anal. Math. Phys., 2(2) (2012), 123–147.

    Article  MATH  MathSciNet  Google Scholar 

  19. L. Hörmander, Hypo-elliptic second order differential equations, Acta Math., 119 (1967), 147–171.

    Article  MATH  MathSciNet  Google Scholar 

  20. I. Cardoso and L. Saal, Explicit fundamental solutions of some second order differential operators on Heisenberg groups, Colloq. Math., 129(2) (2012), 263–288.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc., 258 (1980), 147–153.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Kaplan, On the geometry of groups of Heisenberg type, Bull. Lond. Math. Soc., 15 (1) (1983), 35–42.

    Article  MATH  Google Scholar 

  23. H. B. Lawson and M.-L. Michelson, Spin geometry, Princeton University Press, 1989.

    MATH  Google Scholar 

  24. A. I. Malćev, On a class of homogeneous spaces, Amer. Math. Soc. Translation, 39 (1951); Izv. Akad. Nauk USSR, Ser. Mat., 13 (1949), 9-32.

  25. L. Magnin, Sur les algèbres de Lie nilpotentes de dimension = 7, J. Geom. Phys., 3 (1986), 119–144.

    Article  MATH  MathSciNet  Google Scholar 

  26. C. Seeley, 7 dimensional nilpotent Lie algebras, Trans. Amer. Math. Soc., 335 (1993), 479–496.

    MATH  MathSciNet  Google Scholar 

  27. R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surv. Monogr., 91, Am. Math. Soc., Providence RI, 2002.

  28. M. S. Raghunathan, Discrete subgroups of Lie groups, Springer, New York -Heidelberg -Berlin, 1972.

    Book  MATH  Google Scholar 

  29. P. K. Rashevskii, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski ped. inst. Libknexta, 2 (1938), 83–94 (in Russian).

    Google Scholar 

  30. J. Tie, The inverse of some differential operators on the Heisenberg group, Comm. Partial Differential Equations, 20(7&8) (1995), 1275–1302.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Analysis, Leibniz Universität, Welfengarten 1, 30167, Hannover, Germany

    Wolfram Bauer

  2. Department of Mathematics, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan

    Kenro Furutani

  3. Department of Mathematics, University of Hyogo, 2167 Shosha Himeji, Hyogo, 671-2201, Japan

    Chisato Iwasaki

Authors
  1. Wolfram Bauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenro Furutani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chisato Iwasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wolfram Bauer.

Additional information

Dedicated to Professor Kalyan B. Sinha on the occasion of his 70th birthday.

The first named author has been supported by an Emmy-Noether grant of Deutsche Forschungsgemeinschaft (DFG); the second named author was supported by a fund of Prof. Li Yutian, Hong Kong Baptist University and the National Center for Theoretical Sciences, National Tsing Hua University, Taiwan. The third named author was supported by the Grant-in-aid for Scientific Research (C) No. 24540189, JSPS and the National Center for Theoretical Sciences, National Tsing Hua University, Taiwan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauer, W., Furutani, K. & Iwasaki, C. Spectral zeta function on pseudo H-type nilmanifolds. Indian J Pure Appl Math 46, 539–582 (2015). https://doi.org/10.1007/s13226-015-0151-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-015-0151-6

Keywords

Search

Navigation