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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202101.png" /> be a Riemannian [[Submersion|submersion]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202103.png" /> be operators of Laplace type (cf. also [[Laplace operator|Laplace operator]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202105.png" /> on bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202107.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s1202109.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021012.png" />. One wants to have examples where there exists
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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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
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\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
 
One also wants to know when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \pi ^ { * } E ( \lambda , D _ { Y } ) \subset E ( \mu ( \lambda ) , D _ { Z } ). \end{equation}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021015.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021016.png" /> be the volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021017.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021018.png" /> be the Laplace–Beltrami operator (cf. also [[Laplace–Beltrami equation|Laplace–Beltrami equation]]). Y. Muto [[#References|[a8]]], [[#References|[a7]]] observed that
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021019.png" /></td> </tr></table>
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\begin{equation*} 0 \neq \nu _ { 2 } \in E ( 0 , \Delta _ { S^2 } ^ { 2 } ) \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021020.png" /></td> </tr></table>
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\begin{equation*} \pi ^ { * } \nu _ { 2 } \in E ( \mu , \Delta _ { S^3 } ^ { 2 } ) \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021021.png" />; he also gave other examples involving principal fibre bundles.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021023.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021024.png" /> (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]]].
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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====
 
====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,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120210/s12021025.png" /> commuting mappings and Betti numbers"  ''Tôhoku Math. J.'' , '''27'''  (1975)  pp. 135–152</TD></TR></table>
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<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>

Revision as of 17:02, 1 July 2020

Let $\pi : Z \rightarrow Y$ be a Riemannian submersion. Let $D _ { Y }$ and $D _ { Z }$ be operators of Laplace type (cf. also 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 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). Y. Muto [a8], [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 [a2] and B. Watson [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 [a5] for the real Laplacian and in [a3] for the complex Laplacian. If (a1) holds for a single eigenvalue, then $\lambda \leq \mu$ (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, "$\delta$ commuting mappings and Betti numbers" Tôhoku Math. J. , 27 (1975) pp. 135–152
How to Cite This Entry:
Spectral geometry of Riemannian submersions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_geometry_of_Riemannian_submersions&oldid=50441
This article was adapted from an original article by J.H. Park (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article