### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 091, 16 pages      arXiv:2009.03276      https://doi.org/10.3842/SIGMA.2022.091

### Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces

Akira Kitaoka
Graduate School of Mathematical Sciences, The University of Tokyo, Japan

Received May 19, 2022, in final form November 14, 2022; Published online November 28, 2022

Abstract
We express explicitly the analytic torsion functions associated with the Rumin complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. Moreover, we have a formula between this torsion and the Ray-Singer torsion.

Key words: analytic torsion; Rumin complex; CR geometry; contact geometry.

pdf (394 kb)   tex (19 kb)

References

1. Albin P., Quan H., Sub-Riemannian limit of the differential form heat kernels of contact manifolds, Int. Math. Res. Not. 2022 (2022), 5818-5881, arXiv:1912.02326.
2. Bismut J.M., Zhang W., An extension of a theorem by Cheeger and Müller (with an appendix by François Laudenbach), Astérisque 205 (1992), 235 pages.
3. de Melo T., Spreafico M., Reidemeister torsion and analytic torsion of spheres, J. Homotopy Relat. Struct. 4 (2009), 181-185, arXiv:0906.2570.
4. Fulton W., Young tableaux: with applications to representation theory and geometry, London Math. Soc. Stud. Texts, Vol. 35, Cambridge University Press, Cambridge, 1996.
5. Fulton W., Harris J., Representation theory. A first course, Grad. Texts in Math., Vol. 129, Springer-Verlag, New York, 1991.
6. Goodman R., Wallach N.R., Symmetry, representations, and invariants, Grad. Texts in Math., Vol. 255, Springer, Dordrecht, 2009.
7. Julg P., Kasparov G., Operator $K$-theory for the group ${\rm SU}(n,1)$, J. Reine Angew. Math. 463 (1995), 99-152.
8. Kitaoka A., Analytic torsions associated with the Rumin complex on contact spheres, Internat. J. Math. 31 (2020), 2050112, 16 pages, arXiv:1911.03092.
9. Ponge R., Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry, J. Funct. Anal. 252 (2007), 399-463, arXiv:math.DG/0607296.
10. Ray D.B., Reidemeister torsion and the Laplacian on lens spaces, Adv. Math. 4 (1970), 109-126.
11. Rosenberg S., The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds, London Math. Soc. Stud. Texts, Vol. 31, Cambridge University Press, 1997.
12. Rumin M., Formes différentielles sur les variétés de contact, J. Differential Geom. 39 (1994), 281-330.
13. Rumin M., Sub-Riemannian limit of the differential form spectrum of contact manifolds, Geom. Funct. Anal. 10 (2000), 407-452.
14. Rumin M., Seshadri N., Analytic torsions on contact manifolds, Ann. Inst. Fourier (Grenoble) 62 (2012), 727-782, arXiv:0802.0123.
15. Weng L., You Y., Analytic torsions of spheres, Internat. J. Math. 7 (1996), 109-125.