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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201001.png" /> be the Riemann [[Curvature tensor|curvature tensor]] of a [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201002.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201003.png" /> is an orthonormal basis for an oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201004.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201005.png" /> in the tangent space at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201006.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201007.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201008.png" /> be the skew-symmetric curvature operator introduced by R. Ivanova and G. Stanilov [[#References|[a3]]]. The Riemannian metric is said to be an Ivanov–Petrova metric if the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i1201009.png" /> depend only on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010010.png" /> but not upon the particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010011.png" />-plane in question.
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Let $R$ be the Riemann [[Curvature tensor|curvature tensor]] of a [[Riemannian manifold|Riemannian manifold]] $( M , g )$. If $\{ X , Y \}$ is an orthonormal basis for an oriented $2$-plane $\pi$ in the tangent space at a point $P$ of $M$, let $R ( \pi ) = R ( X , Y )$ be the skew-symmetric curvature operator introduced by R. Ivanova and G. Stanilov [[#References|[a3]]]. The Riemannian metric is said to be an Ivanov–Petrova metric if the eigenvalues of $R ( \pi )$ depend only on the point $P$ but not upon the particular $2$-plane in question.
  
 
===Example 1.===
 
===Example 1.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010012.png" /> is a metric of constant sectional curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010013.png" />, then the group of local isometries acts transitively on the Grassmannian of oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010014.png" />-planes and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010015.png" /> is Ivanov–Petrova. The eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010016.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010017.png" />.
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If $g$ is a metric of constant sectional curvature $C$, then the group of local isometries acts transitively on the Grassmannian of oriented $2$-planes and hence $( M , g )$ is Ivanov–Petrova. The eigenvalues of $R ( \pi )$ are $\{ \pm i C , 0 , \ldots , 0 \}$.
  
 
===Example 2.===
 
===Example 2.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010018.png" /> be a product manifold, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010019.png" /> is a subinterval of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010020.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010021.png" /> is a metric of constant sectional curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010023.png" />. Give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010024.png" /> the metric
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Let $M = I \times N$ be a product manifold, where $I$ is a subinterval of $\mathbf{R}$ and where $d s _ { N } ^ { 2 }$ is a metric of constant sectional curvature $K$ on $N$. Give $M$ the metric
  
<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/i/i120/i120100/i12010025.png" /></td> </tr></table>
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\begin{equation*} d s _ { M } ^ { 2 } = d t ^ { 2 } + f ( t ) d s _ { N } ^ { 2 }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010026.png" />. One can then compute that the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010027.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010029.png" />. Thus, this metric is Ivanov–Petrova.
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where $f ( t ) = ( K t ^ { 2 } + A t + B ) / 2 &gt; 0$. One can then compute that the eigenvalues of $R ( \pi )$ are $\{ \pm i C ( t ) , 0 , \ldots , 0 \}$ for $C ( t ) = ( 4 K B - A ^ { 2 } ) / 4 f ( t ) ^ { 2 }$. Thus, this metric is Ivanov–Petrova.
  
In Example 1, the eigenvalues of the skew-symmetric curvature operator are constant; in Example 2, the eigenvalues depend upon the point of the manifold. S. Ivanov and I. Petrova [[#References|[a2]]] showed that in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010030.png" />, any Riemannian manifold which is Ivanov–Petrova is locally isometric to one of the two metrics exhibited above. This result was later generalized [[#References|[a4]]], [[#References|[a1]]] to dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010032.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010033.png" />; the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010034.png" /> is exceptional and is still open (1998). Partial results in the Lorentzian setting have been obtained by T. Zhang [[#References|[a5]]].
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In Example 1, the eigenvalues of the skew-symmetric curvature operator are constant; in Example 2, the eigenvalues depend upon the point of the manifold. S. Ivanov and I. Petrova [[#References|[a2]]] showed that in dimension $m = 4$, any Riemannian manifold which is Ivanov–Petrova is locally isometric to one of the two metrics exhibited above. This result was later generalized [[#References|[a4]]], [[#References|[a1]]] to dimensions $m = 5$, $m = 6$, and $m \geq 8$; the case $m = 7$ is exceptional and is still open (1998). Partial results in the Lorentzian setting have been obtained by T. Zhang [[#References|[a5]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010035.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010036.png" />-tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010037.png" /> which defines a corresponding curvature operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010039.png" /> satisfies the identities,
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Let $R ( X , Y , Z , W )$ be a $4$-tensor on $\mathbf{R} ^ { m }$ which defines a corresponding curvature operator $R ( X , Y )$. If $R$ satisfies the identities,
  
<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/i/i120/i120100/i12010040.png" /></td> </tr></table>
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\begin{equation*} R ( X , Y ) = - R ( Y , X ), \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/i/i120/i120100/i12010041.png" /></td> </tr></table>
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\begin{equation*} g ( R ( X , Y ) Z , W ) = g ( R ( Z , W ) X , Y ) , R ( X , Y ) Z + R ( Y , Z ) X + R ( Z , X ) Y = 0, \end{equation*}
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010042.png" /> is said to be an algebraic curvature tensor. The algebraic curvature tensors which are Ivanov–Petrova have also been classified; they are known to have rank at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010043.png" /> in all dimensions except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010045.png" />, and have the form
+
then $R$ is said to be an algebraic curvature tensor. The algebraic curvature tensors which are Ivanov–Petrova have also been classified; they are known to have rank at most $2$ in all dimensions except $m = 4$ and $m = 7$, and have the form
  
<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/i/i120/i120100/i12010046.png" /></td> </tr></table>
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\begin{equation*} R ( X , Y ) Z = C \{ g ( \phi Y , Z ) \phi X - g ( \phi X , Z ) \phi Y \}, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010047.png" /> is an isometry with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010048.png" />. Note that in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010049.png" />, there is an algebraic curvature tensor which is Ivanov–Petrova, has rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010050.png" /> and which is constructed using the quaternions; up to scaling and change of basis it is unique and the non-zero entries (up to the usual curvature symmetries) are given by:
+
where $\phi$ is an isometry with $\phi ^ { 2 } = \operatorname{id}$. Note that in dimension $m = 4$, there is an algebraic curvature tensor which is Ivanov–Petrova, has rank $4$ and which is constructed using the quaternions; up to scaling and change of basis it is unique and the non-zero entries (up to the usual curvature symmetries) are given by:
  
<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/i/i120/i120100/i12010051.png" /></td> </tr></table>
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\begin{equation*} R _ { 1212 } = a _ { 2 } , R _ { 1313 } = a _ { 2 } , R _ { 2424 } = a _ { 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/i/i120/i120100/i12010052.png" /></td> </tr></table>
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\begin{equation*} R _ { 1414 } = a _ { 1 } , R _ { 2323 } = a _ { 1 } , R _ { 3434 } = a _ { 2 } , R _ { 1234 } = a _ { 1 } , R _ { 1324 } = - a _ { 1 } , R _ { 1423 } = a _ { 2 }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010053.png" />. The situation in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i120/i120100/i12010054.png" /> is open (1998).
+
where $a _ { 2 } + 2 a_ { 1 } = 0$. The situation in dimension $m = 7$ is open (1998).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Gilkey,  J.V. Leahy,  H. Sadofsky,  "Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues"  ''Indiana J.''  (to appear)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Ivanov,  I. Petrova,  "Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues"  ''Geom. Dedicata'' , '''70'''  (1998)  pp. 269–282</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Ivanova,  G. Stanilov,  "A skew-symmetric curvature operator in Riemannian geometry"  M. Behara (ed.)  R. Fritsch (ed.)  R. Lintz (ed.) , ''Symposia Gaussiana, Conf. A''  (1995)  pp. 391–395</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Gilkey,  "Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues II" , ''Proc. Diff. Geom. Symp. (Brno, 1998)''  (to appear)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Zhang,  "Manifolds with indefinite metrics whose skew symmetric curvature operator has constant eigenvalues"  ''PhD Thesis Univ. Oregon''  (2000)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Gilkey,  J.V. Leahy,  H. Sadofsky,  "Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues"  ''Indiana J.''  (to appear)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Ivanov,  I. Petrova,  "Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues"  ''Geom. Dedicata'' , '''70'''  (1998)  pp. 269–282</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R. Ivanova,  G. Stanilov,  "A skew-symmetric curvature operator in Riemannian geometry"  M. Behara (ed.)  R. Fritsch (ed.)  R. Lintz (ed.) , ''Symposia Gaussiana, Conf. A''  (1995)  pp. 391–395</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Gilkey,  "Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues II" , ''Proc. Diff. Geom. Symp. (Brno, 1998)''  (to appear)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  T. Zhang,  "Manifolds with indefinite metrics whose skew symmetric curvature operator has constant eigenvalues"  ''PhD Thesis Univ. Oregon''  (2000)</td></tr></table>

Latest revision as of 16:59, 1 July 2020

Let $R$ be the Riemann curvature tensor of a Riemannian manifold $( M , g )$. If $\{ X , Y \}$ is an orthonormal basis for an oriented $2$-plane $\pi$ in the tangent space at a point $P$ of $M$, let $R ( \pi ) = R ( X , Y )$ be the skew-symmetric curvature operator introduced by R. Ivanova and G. Stanilov [a3]. The Riemannian metric is said to be an Ivanov–Petrova metric if the eigenvalues of $R ( \pi )$ depend only on the point $P$ but not upon the particular $2$-plane in question.

Example 1.

If $g$ is a metric of constant sectional curvature $C$, then the group of local isometries acts transitively on the Grassmannian of oriented $2$-planes and hence $( M , g )$ is Ivanov–Petrova. The eigenvalues of $R ( \pi )$ are $\{ \pm i C , 0 , \ldots , 0 \}$.

Example 2.

Let $M = I \times N$ be a product manifold, where $I$ is a subinterval of $\mathbf{R}$ and where $d s _ { N } ^ { 2 }$ is a metric of constant sectional curvature $K$ on $N$. Give $M$ the metric

\begin{equation*} d s _ { M } ^ { 2 } = d t ^ { 2 } + f ( t ) d s _ { N } ^ { 2 }, \end{equation*}

where $f ( t ) = ( K t ^ { 2 } + A t + B ) / 2 > 0$. One can then compute that the eigenvalues of $R ( \pi )$ are $\{ \pm i C ( t ) , 0 , \ldots , 0 \}$ for $C ( t ) = ( 4 K B - A ^ { 2 } ) / 4 f ( t ) ^ { 2 }$. Thus, this metric is Ivanov–Petrova.

In Example 1, the eigenvalues of the skew-symmetric curvature operator are constant; in Example 2, the eigenvalues depend upon the point of the manifold. S. Ivanov and I. Petrova [a2] showed that in dimension $m = 4$, any Riemannian manifold which is Ivanov–Petrova is locally isometric to one of the two metrics exhibited above. This result was later generalized [a4], [a1] to dimensions $m = 5$, $m = 6$, and $m \geq 8$; the case $m = 7$ is exceptional and is still open (1998). Partial results in the Lorentzian setting have been obtained by T. Zhang [a5].

Let $R ( X , Y , Z , W )$ be a $4$-tensor on $\mathbf{R} ^ { m }$ which defines a corresponding curvature operator $R ( X , Y )$. If $R$ satisfies the identities,

\begin{equation*} R ( X , Y ) = - R ( Y , X ), \end{equation*}

\begin{equation*} g ( R ( X , Y ) Z , W ) = g ( R ( Z , W ) X , Y ) , R ( X , Y ) Z + R ( Y , Z ) X + R ( Z , X ) Y = 0, \end{equation*}

then $R$ is said to be an algebraic curvature tensor. The algebraic curvature tensors which are Ivanov–Petrova have also been classified; they are known to have rank at most $2$ in all dimensions except $m = 4$ and $m = 7$, and have the form

\begin{equation*} R ( X , Y ) Z = C \{ g ( \phi Y , Z ) \phi X - g ( \phi X , Z ) \phi Y \}, \end{equation*}

where $\phi$ is an isometry with $\phi ^ { 2 } = \operatorname{id}$. Note that in dimension $m = 4$, there is an algebraic curvature tensor which is Ivanov–Petrova, has rank $4$ and which is constructed using the quaternions; up to scaling and change of basis it is unique and the non-zero entries (up to the usual curvature symmetries) are given by:

\begin{equation*} R _ { 1212 } = a _ { 2 } , R _ { 1313 } = a _ { 2 } , R _ { 2424 } = a _ { 2 }, \end{equation*}

\begin{equation*} R _ { 1414 } = a _ { 1 } , R _ { 2323 } = a _ { 1 } , R _ { 3434 } = a _ { 2 } , R _ { 1234 } = a _ { 1 } , R _ { 1324 } = - a _ { 1 } , R _ { 1423 } = a _ { 2 }, \end{equation*}

where $a _ { 2 } + 2 a_ { 1 } = 0$. The situation in dimension $m = 7$ is open (1998).

References

[a1] P. Gilkey, J.V. Leahy, H. Sadofsky, "Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues" Indiana J. (to appear)
[a2] S. Ivanov, I. Petrova, "Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues" Geom. Dedicata , 70 (1998) pp. 269–282
[a3] R. Ivanova, G. Stanilov, "A skew-symmetric curvature operator in Riemannian geometry" M. Behara (ed.) R. Fritsch (ed.) R. Lintz (ed.) , Symposia Gaussiana, Conf. A (1995) pp. 391–395
[a4] P. Gilkey, "Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues II" , Proc. Diff. Geom. Symp. (Brno, 1998) (to appear)
[a5] T. Zhang, "Manifolds with indefinite metrics whose skew symmetric curvature operator has constant eigenvalues" PhD Thesis Univ. Oregon (2000)
How to Cite This Entry:
Ivanov-Petrova metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ivanov-Petrova_metric&oldid=22605
This article was adapted from an original article by P.B. Gilkey (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article