Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues (1900–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
Request Permissions
 
 

 

Curvature invariants, differential operators
and local homogeneity


Authors: Friedbert Prüfer, Franco Tricerri and Lieven Vanhecke
Journal: Trans. Amer. Math. Soc. 348 (1996), 4643-4652
MSC (1991): Primary 53C20, 53C25, 53C30
DOI: https://doi.org/10.1090/S0002-9947-96-01686-8
MathSciNet review: 1363946
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We first prove that a Riemannian manifold $(M,g)$ with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold $(M,g)$ whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

References [Enhancements On Off] (What's this?)

  • 1. M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Math. 194, Springer-Verlag, Berlin, Heidelberg, New York, 1971. MR 43:8025
  • 2. J. Berndt, F. Tricerri and L. Vanhecke, Geometry of generalized Heisenberg groups and their Damek-Ricci harmonic extensions, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 471-476. MR 95a:53074
  • 3. J. Berndt, F. Tricerri and L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Math. 1598, Springer-Verlag, Berlin, Heidelberg, New York, 1995. CMP 1995:15
  • 4. J. Berndt and L. Vanhecke, Two natural generalizations of locally symmetric spaces, Diff. Geom. Appl. 2 (1992), 57-80. MR 94j:53061
  • 5. E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 139-142. MR 93b:53043
  • 6. J. E. D'Atri and H. K. Nickerson, Divergence-preserving geodesic symmetries, J. Differential Geom. 3 (1969), 467-476. MR 41:7574
  • 7. A. Gray and T. J. Willmore, Mean-value theorems for Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), 343-364. MR 84f:53038
  • 8. M. Gromov, Partial Differential Relations, Ergeb. Math. Grenzgeb. 3. Folge 9, Springer-Verlag, Berlin, Heidelberg, New York, 1986. MR 90a:58201
  • 9. O. Kowalski, Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale Settembre, 1983, pp. 223-232.
  • 10. O. Kowalski, Counter-example to the ``Second Singer's Theorem", Ann. Global Anal. Geom. 8 (1990), 211-214. MR 92b:53057
  • 11. O. Kowalski, F. Prüfer and L. Vanhecke, D'Atri spaces, preprint, 1995.
  • 12. O. Kowalski and L. Vanhecke, Ball-homogeneous and disk-homogeneous Riemannian manifolds, Math. Z. 180 (1982), 429-444. MR 84c:53043
  • 13. L. Nicolodi and F. Tricerri, On two theorems of I.M. Singer about homogeneous spaces, Ann. Global Anal. Geom. 8 (1990), 193-209. MR 92b:53075
  • 14. V. Poenaru, Singularités $C^{\infty }$ en présence de symétrie, Lecture Notes in Math. 510, Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 55:13471
  • 15. C. Procesi and G. Schwarz, Inequalities defining orbit spaces, Inv. Math. 81 (1985), 539-554. MR 87h:20078
  • 16. F. Prüfer, On compact Riemannian manifolds with volume-preserving symmetries, Ann. Global Anal. Geom. 7 (1989), 133-140. MR 91c:53047
  • 17. I.M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697. MR 24:A1100
  • 18. A. Spiro, Lie pseudogroups and locally homogeneous Riemannian spaces, Boll. Un. Mat. Ital. B 6 (1992), 843-872. MR 94a:53085
  • 19. T. Sumitomo, On the commutator of differential operators, Hokkaido Math. J. 1 (1972), 30-42. MR 47:5785
  • 20. Z.I. Szabó, Spectral theory for operator families on Riemannian manifolds, Differential geometry, Proc. Summer Research Institute on Differential Geometry, University of California, Los Angeles, 1990, Proc. Sympos. Pure Math., 54, Part 3 (R. Greene and S.T. Yau, eds.), 1993, pp. 615-665. MR 94k:58152
  • 21. F. Tricerri, Locally homogeneous Riemannian manifolds, Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), 411-426. MR 94k:53065
  • 22. L. Vanhecke, Geometry in normal and tubular neighborhoods, Proc. Workshop on Differential Geometry and Topology Cala Gonone (Sardinia) 1988, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al vol. 58 (1988), 73-176. MR 92g:53037
  • 23. L. Vanhecke and T. J. Willmore, Interaction of tubes and spheres, Math. Ann. 263 (1983), 31-42. MR 85c:53085
  • 24. H. Weyl, The classical groups, their invariants and representations, Princeton Univ. Press, 1946. MR 1:42c

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C20, 53C25, 53C30

Retrieve articles in all journals with MSC (1991): 53C20, 53C25, 53C30

Additional Information

Friedbert Prüfer
Affiliation: Universität Leipzig, Fakultät für Mathematik und Informatik, Mathematisches Institut, Augustusplatz 10, D-04109, Leipzig, Germany
Email: pruefer@mathematik.uni-leipzig.d400.de

Franco Tricerri
Affiliation: Universität Leipzig, Fakultät für Mathematik und Informatik, Mathematisches Institut, Augustusplatz 10, D-04109, Leipzig, Germany

Lieven Vanhecke
Affiliation: Katholieke Universiteit Leuven, Departement of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: lieven.vanhecke@wis.kuleuven.ac.be

DOI: https://doi.org/10.1090/S0002-9947-96-01686-8
Keywords: Curvature invariants, locally homogeneous spaces, Laplacian, invariant differential operators, commutativity, spaces with volume-preserving geodesic symmetries
Received by editor(s): September 26, 1995
Additional Notes: $^{†}$To our deep sorrow F. Tricceri died in an airplane crash in China on the sixth of June 1995. His contribution to this paper was essential.
Article copyright: © Copyright 1996 American Mathematical Society