Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지

ON THE ISOSPECTRA AND THE ISOMETRIES OF THE ALOFF-WALLACH SPACES

  • Joe, Do-Sang (DEPARTMENT OF MATHEMATICS, EWHA WOMAN\'S UNIVERSITY) ;
  • Lee, Yoon-Weon (DEPARTMENT OF MATHEMATICS, INHA UNIVERSITY) ;
  • Park, Jin-Sung (DEPARTMENT OF MATHEMATICS, KOREA INSTITUTE FOR ADVANCED STUDY) ;
  • Ryu, Jeong-Seog (DEPARTMENT OF MATHEMATICS EDUCATION, COLLEGE OF EDUCATION, HONGIK UNIVERSITY)
  • Published : 2001.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

We use the branching rules on SU(3) to show that if two Aloff-Wallach spaces $M_{k,l}\;and\;M_{k',l'}$ are isospectral for the Laplacian acting on smooth functions, they are isometric. We also show that 1 is the non-zero smallest eigenvalue among all Aloff-Wallach spaces and compute the multiplicities.

Keywords

References

  1. Bull. Amer. Math. Soc. v.81 An infinite family of distinct 7-manifolds admitting positively curved riemannian structures S. Aloff;N. R. Wallach
  2. Arch. Math. v.59 The Dirac operator on homogeneous spaces and its spectrum on 3=dimensional lens spaces C. Bar
  3. Introduction to Compact Lie Groups, Series in Pure Mathematics v.13 H. D. Fegan
  4. J. Math. v.15 Spectra and eigenforms of Laplacian on S and P C A. Ikeda;Y. Taniguchi
  5. Math. Ann. v.297 Equivariant analytic torsion on P C K. Kohler
  6. Ann. Math. v.127 A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with SU(3) X SU(2) X U(1)-symmetry M. Kreck;S. Stolz
  7. Jour. Diff. Geom v.33 Some nondiffiomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature
  8. Trans.of A.M.S. v.349 no.7 A homotopy classification of certain 7-manifolds B. Kuggel
  9. Ann. Global Anal. and Geom v.17 The spectrum of the Dirac operator on G₂/So(4) L. Seeger