View source for Spectral geometry of Riemannian submersions

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Let $\pi : Z \rightarrow Y$ be a Riemannian [[Submersion|submersion]]. Let $D _ { Y }$ and $D _ { Z }$ be operators of Laplace type (cf. also [[Laplace operator|Laplace operator]]) on $Y$ and $Z$ on bundles $V _ { Y }$ and $V _ { Z }$. Let $E ( \lambda , D _ { Y } )$ and $E ( \lambda , D _ { \operatorname {Z} } )$ be the corresponding eigenspaces. Assume given a pull-back <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021010.png"/> from $V _ { Y }$ to $V _ { Z }$. One wants to have examples where there exists

\begin{equation} \tag{a1} 0 \neq \phi \in E ( \lambda , D _ { Y } ) \text { with } \pi ^ { * } \phi \in E ( \mu , D _ { Z } ). \end{equation}

One also wants to know when

\begin{equation} \tag{a2} \pi ^ { * } E ( \lambda , D _ { Y } ) \subset E ( \mu ( \lambda ) , D _ { Z } ). \end{equation}

Let $\pi : S ^ { 3 } \rightarrow S ^ { 2 }$ and let $\nu_2$ be the volume element on $S ^ { 2 }$. Let $\Delta ^ { p }$ be the Laplace–Beltrami operator (cf. also [[Laplace–Beltrami equation|Laplace–Beltrami equation]]). Y. Muto [[#References|[a8]]], [[#References|[a7]]] observed that

\begin{equation*} 0 \neq \nu _ { 2 } \in E ( 0 , \Delta _ { S^2 } ^ { 2 } ) \end{equation*}

\begin{equation*} \pi ^ { * } \nu _ { 2 } \in E ( \mu , \Delta _ { S^3 } ^ { 2 } ) \end{equation*}

for $\mu \neq 0$; he also gave other examples involving principal fibre bundles.

S.I. Goldberg and T. Ishihara [[#References|[a2]]] and B. Watson [[#References|[a9]]] studied this question and determined some conditions to ensure that (a2) holds with $\mu ( \lambda ) = \lambda$ for all $\lambda$; this work was later extended in [[#References|[a5]]] for the real Laplacian and in [[#References|[a3]]] for the complex Laplacian. If (a1) holds for a single eigenvalue, then $\lambda \leq \mu$ (eigenvalues cannot decrease). See also [[#References|[a1]]] for a discussion of the case in which the fibres are totally geodesic. See [[#References|[a6]]] for related results in the spin setting. For a survey of the field, see [[#References|[a4]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  L. Berard Bergery,   J.P. Bourguignon,   "Laplacians and Riemannian submersions with totally geodesic fibers"  ''Illinois J. Math.'' , '''26'''  (1982)  pp. 181–200</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S.I. Goldberg,   T. Ishihara,   "Riemannian submersions commuting with the Laplacian"  ''J. Diff. Geom.'' , '''13'''  (1978)  pp. 139–144</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P. Gilkey,   J. Leahy,   J.H. Park,   "The eigenforms of the complex Laplacian for a holomorphic Hermitian submersion"  ''Nagoya Math. J.''  (to appear)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  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)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Gilkey,   J.H. Park,   "Riemannian submersions which preserve the eigenforms of the Laplacian"  ''Illinois J. Math.'' , '''40'''  (1996)  pp. 194–201</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Moroianu,   "Opérateur de Dirac et Submersions Riemanniennes"  ''Thesis École Polytechn. Palaiseau''  (1996)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  Y. Muto,   "Riemannian submersion and the Laplace–Beltrami operator"  ''Kodai Math. J.'' , '''1'''  (1978)  pp. 329–338</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  Y. Muto,   "Some eigenforms of the Laplace–Beltrami operators in a Riemannian submersion"  ''J. Korean Math. Soc.'' , '''15'''  (1978)  pp. 39–57</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  B. Watson,   "Manifold maps commuting with the Laplacian"  ''J. Diff. Geom.'' , '''8'''  (1973)  pp. 85–94</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  Y. Muto,   "$\delta$ commuting mappings and Betti numbers"  ''Tôhoku Math. J.'' , '''27'''  (1975)  pp. 135–152</td></tr></table>