Abstract
We prove an Atiyah–Bott–Berline–Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern–Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. A. Álvarez López and Y. A. Kordyukov, Lefschetz distributions of Lie foliations, in C*-algebras and Elliptic Theory II, Trends in Mathematics, Birkhäuser, Basel, 2008, pp. 1–40.
J. A. Álvarez López and G. Hector, The dimension of the leafwise reduced cohomology, American Journal of Mathematics, 123 (2001), 607–646.
J. A. Álvarez López and X. Masa, Morphisms between complete Riemannian pseudogroups, Topology and its Applications, 155 (2008), 544–604.
T. Asuke, Residues of the Bott class and an application to the Futaki invariant, Asian Journal of Mathematics, 7 (2003), 239–268.
T. Asuke, Localization and residue of the Bott class, Topology, 43 (2004), 289–317.
T. Asuke, Godbillon–Vey Class of Transversely Holomorphic Foliations, MSJ Memoirs, Vol. 24, Mathematical Society of Japan, Tokyo, 2010.
M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology, 23 (1984), 1–28.
A. Banyaga and P. Rukimbira, On characteristics of circle invariant presymplectic forms, Proceedings of the American Mathematical Society, 123 (1995), 3901–3906.
P. Baum and R. Bott, Singularities of holomorphic foliations, Journal of Differential Geometry, 7 (1972), 279–342.
N. Berline and M. Vergne, Zéros d’un champ de vecteurs et classes caractéristiques équivariantes, Duke Mathematical Journal 50 (1983), 539–549.
N. Berline and M. Vergne, Fourier transforms of orbits of the coadjoint representation, in Representation Theory of Reductive Groups (Park City, Utah, 1982), Progress in Mathematics, Vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 53–67.
R. Bott, Vector fields and characteristic numbers, Michigan Mathematical Journal, 14 (1967), 231–244.
R. Bott, Lectures on characteristic classes and foliations, in Lectures on Algebraic and Differential Topology (Second Latin American School in Math., Mexico City, 1971), Lecture Notes in Mathematics, Vol. 279, Springer, Berlin, 1972, pp. 1–94.
C. P. Boyer and K. Galicki, A note on toric contact geometry, Journal of Geometry and Physics 35 (2000), 288–298.
C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
J. Cantwell and L. Conlon, The dynamics of open, foliated manifolds and a vanishing theorem for the Godbillon–Vey class, Advances in Mathematics, 53 (1984), 1–27.
H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, in Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, pp. 57–71.
L. Casselmann and J. M. Fisher, Localization for K-contact manifolds, Journal of Symplectic Geometry, to appear, arXiv: 1703.00333.
S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals of Mathematics, 99 (1974), 48–69.
D. Domínguez, Finiteness and tenseness theorems for Riemannian foliations, American Journal of Mathematics, 120 (1998), 1237–1276.
J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Inventiones Mathematicae, 69 (1982), 259–268.
G. Duminy, L’invariant de Godbillon–Vey d’un feuilletage se localise dans les feuilles ressort, Preprint (1982).
O. Goertsches, H. Nozawa and D. Töben, Equivariant cohomology of K-contact manifolds, Mathematische Annalen, 354 (2012), 1555–1582.
O. Goertsches and D. Töben, Equivariant basic cohomology of Riemannian foliations, Journal für die Reine und Angewandte Mathematik, to appear, https://doi.org/10.1515/crelle-2015-0102.
W. Greub, S. Halperin and R. Vanstone, Connections, Curvature, and Cohomology. Volumes I–III, Pure and Applied Mathematics, Vol. 47, Academic Press, New York–London, 1972, 1973, 1976.
V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Mathematics Past and Present, Springer, Berlin, 1999.
A. Haefliger, Feuilletages sur les variétés ouvertes, Topology, 9 (1970), 183–194.
A. Haefliger, Homotopy and integrability, in Manifolds—Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer-Verlag, Berlin, 1971, pp. 133–163.
J. L. Heitsch and S. Hurder, Secondary classes, Weil measures and the geometry of foliations, Journal of Differential Geometry, 20 (1984), 291–539.
S. Hurder, Characteristic classes for Riemannian foliations, in Differential Geometry, World Scientific Publishing, Hackensack, NJ, 2009, pp. 11–35.
S. Hurder and A. Katok, Ergodic Theory and Weil Measures for Foliations, Annals of Mathematics, 126 (1987), 221–275.
L. Jeffrey and F. Kirwan, Localization for nonabelian group actions, Topology, 34 (1995), 291–327.
F. Kamber and P. Tondeur, Duality for Riemannian foliations, in Singularities, Part 1 (Arcata, CA, 1981), Proceedings of Symposia in Pure Mathematics, Vol. 40, American Mathematical Society, Providence, RI, 1983, pp. 609–618.
J. Lawrence, Polytope volume computation, Mathematics of Computation, 57 (1991), 259–271.
C. Lazarov and J. Pasternack, Secondary characteristic classes for Riemannian foliations, Journal of Differential Geometry, 11 (1976), 365–385.
E. Lerman, Contact toric manifolds, Journal of Symplectic Geometry, 1 (2002), 785–828.
D. Martelli, J. Sparks and S.-T. Yau, The geometric dual of a-maximisation for toric Sasaki–Einstein manifolds, Communications in Mathematical Physics, 268 (2006), 39–65.
D. Martelli, J. Sparks and S.-T. Yau, Sasaki–Einstein manifolds and volume minimisation, Communications in Mathematical Physics, 280 (2008), 611–673.
X. Masa, Duality and minimality in Riemannian foliations, Commentarii Mathematici Helvetici, 67 (1992), 17–27.
V. Mathai and D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology, 25 (1986), 85–110.
T. Matsuoka and S. Morita, On characteristic classes of Kähler foliations, Osaka Journal of Mathematics, 16 (1979), 539–550.
E. Meinrenken, Equivariant cohomology and the Cartan model, in Encyclopedia of Mathematical Physics, Academic Press, Cambridge, MA, 2006, pp. 242–250.
P. Molino, Riemannian Foliations, Progress in Mathematics, Vol. 73, Birkhäuser, Boston, MA, 1988.
P. Molino and V. Sergiescu, Deux remarques sur les flots Riemanniens, Manuscripta Mathematica, 51 (1985), 145–161.
S. Morita, On characteristic classes of Riemannian foliations, Osaka Journal of Mathematics, 16 (1979), 161–172.
H. Nozawa, Five dimensional K-contact manifolds of rank 2. Doctoral Thesis, University of Tokyo 2009, available at http://arxiv.org/abs/0907.0208
H. Nozawa, Haefliger cohomology of Riemannian foliations, Preprint, (2012), available at http://arxiv.org/abs/1209.3817
H. Nozawa and J. I. Royo Prieto, Tenseness of Riemannian flows, Université de Grenoble. Annales de l’Institut Fourier, 64 (2014), 1419–1439.
B. Reinhart, Harmonic integrals on foliated manifolds, American Journal of Mathematics, 81 (1959), 529–536.
P. Rukimbira, The dimension of leaf closures of K-contact flows, Annals of Global Analysis and Geometry, 12 (1994), 103–108.
K. S. Sarkaria, A finiteness theorem for foliated manifolds, Journal of the Mathematical Society of Japan 30 (1978), 687–696.
V. Sergiescu, Cohomologie basique et dualité des feuilletages riemanniens, Universit é de Grenoble. Annales de l’Institut Fourier, 35 (1985), 137–158.
T. Takahashi, Deformations of Sasakian structures and its application to the Brieskorn manifolds, Tôhoku Mathematical Journal, 30 (1978), 37–43.
D. Töben, Localization of basic characteristic classes, Université de Grenoble. Annales de l’Institut Fourier, 64 (2014), 537–570.
P. Tondeur, Geometry of Foliations, Monographs in Mathematics, Vol. 90, Birkhäuser Verlag, Basel, 1997.
T. Yamazaki, A construction of K-contact manifolds by a fiber join. Tôhoku Mathematical Journal 51 (1999), 433–446.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author is partially supported by Research Fellowship of the Canon Foundation in Europe, the EPDI/JSPS/IHÉS Fellowship, the Spanish MICINN grant MTM2011-25656 and JSPS KAKENHI Grant Number 26800047.
Rights and permissions
About this article
Cite this article
Goertsches, O., Nozawa, H. & Töben, D. Localization of Chern–Simons type invariants of Riemannian foliations. Isr. J. Math. 222, 867–920 (2017). https://doi.org/10.1007/s11856-017-1608-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1608-6