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, " commuting mappings and Betti numbers" Tôhoku Math. J. , 27 (1975) pp. 135–152