Ivanov-Petrova metric

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)