Spectral geometry of Riemannian submersions
Let be a Riemannian submersion. Let
and
be operators of Laplace type (cf. also Laplace operator) on
and
on bundles
and
. Let
and
be the corresponding eigenspaces. Assume given a pull-back
from
to
. One wants to have examples where there exists
![]() | (a1) |
One also wants to know when
![]() | (a2) |
Let and let
be the volume element on
. Let
be the Laplace–Beltrami operator (cf. also Laplace–Beltrami equation). Y. Muto [a8], [a7] observed that
![]() |
![]() |
for ; he also gave other examples involving principal fibre bundles.
S.I. Goldberg and T. Ishihara [a2] and B. Watson [a9] studied this question and determined some conditions to ensure that (a2) holds with for all
; this work was later extended in [a5] for the real Laplacian and in [a3] for the complex Laplacian. If (a1) holds for a single eigenvalue, then
(eigenvalues cannot decrease). See also [a1] for a discussion of the case in which the fibres are totally geodesic. See [a6] for related results in the spin setting. For a survey of the field, see [a4].
References
[a1] | L. Berard Bergery, J.P. Bourguignon, "Laplacians and Riemannian submersions with totally geodesic fibers" Illinois J. Math. , 26 (1982) pp. 181–200 |
[a2] | S.I. Goldberg, T. Ishihara, "Riemannian submersions commuting with the Laplacian" J. Diff. Geom. , 13 (1978) pp. 139–144 |
[a3] | P. Gilkey, J. Leahy, J.H. Park, "The eigenforms of the complex Laplacian for a holomorphic Hermitian submersion" Nagoya Math. J. (to appear) |
[a4] | P. Gilkey, J. Leahy, J.H. Park, "Spinors, spectral geometry, and Riemannian submersions" , Lecture Notes , 40 , Research Inst. Math., Global Analysis Research Center, Seoul Nat. Univ. (1998) |
[a5] | P. Gilkey, J.H. Park, "Riemannian submersions which preserve the eigenforms of the Laplacian" Illinois J. Math. , 40 (1996) pp. 194–201 |
[a6] | A. Moroianu, "Opérateur de Dirac et Submersions Riemanniennes" Thesis École Polytechn. Palaiseau (1996) |
[a7] | Y. Muto, "Riemannian submersion and the Laplace–Beltrami operator" Kodai Math. J. , 1 (1978) pp. 329–338 |
[a8] | Y. Muto, "Some eigenforms of the Laplace–Beltrami operators in a Riemannian submersion" J. Korean Math. Soc. , 15 (1978) pp. 39–57 |
[a9] | B. Watson, "Manifold maps commuting with the Laplacian" J. Diff. Geom. , 8 (1973) pp. 85–94 |
[a10] | Y. Muto, "![]() |
Spectral geometry of Riemannian submersions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_geometry_of_Riemannian_submersions&oldid=18971