Osserman conjecture

Let $R$ be the Riemann curvature tensor of a Riemannian manifold $(M,g)$. Let $J(X):Y\to R(Y,X)X)$ be the Jacobi operator. If $X$ is a unit tangent vector at a point $P$ of $M$, then $J(X)$ is a self-adjoint endomorphism of the tangent bundle at $P$. If $(M,g)$ is flat or is locally a rank-$1$ symmetric space (cf. also Symmetric space), then the set of local isometries acts transitively on the sphere bundle $S(TM)$ of unit tangent vectors, so $J(X)$ has constant eigenvalues on $S(TM)$. R. Osserman [a6] wondered if the converse implication was valid; the following conjecture has become known as the Osserman conjecture: If $J(X)$ has constant eigenvalues, then $(M,g)$ is flat or is locally a rank-$1$ symmetric space.

Let $m$ be the dimension of $M$. If $m$ is odd, if $m\equiv2$ modulo $4$, or if $m=4$, then C.S. Chi [a3] has established this conjecture using a blend of tools from algebraic topology and differential geometry. There is a corresponding purely algebraic problem. Let $R(X,Y,Z,W)$ be a $4$-tensor on $R^m$ which defines a corresponding curvature operator $R(X,Y)$. If $R$ satisfies the identities,

\begin{equation}R(X,Y)=-R(Y,X),\\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 Riemann curvature tensor of a Riemannian metric is an algebraic curvature tensor. Conversely, given an algebraic curvature tensor at a point $P$ of $M$, there always exists a Riemannian metric whose curvature tensor at $P$ is $R$. Let $J(X):Y\to R(Y,X)X$; this is a self-adjoint endomorphism of the tangent bundle at $P$. One says that $R$ is Osserman if the eigenvalues of $J(X)$ are constant on the unit sphere $S^{m-1}$ in $R^m$. C.S. Chi classified the Osserman algebraic curvature tensors for $m$ odd or $m\equiv 2$ modulo $4$; he then used the second Bianchi identity to complete the proof. However, if $m\equiv 0$ modulo $4$, it is known [a4] that there are Osserman algebraic curvature tensors which are not the curvature tensors of rank-$1$ symmetric spaces and the classification promises to be considerably more complicated in these dimensions.

There is a generalization of this conjecture to metrics of higher signature. In the Lorentzian setting, one can show that any algebraic curvature tensor which is Osserman is the algebraic curvature tensor of a metric of constant sectional curvature; it then follows that any Osserman Lorentzian metric has constant sectional curvature [a2]. For metrics of higher signature, the Jordan normal form of the Jacobi operator enters; the Jacobi operator need not be diagonalizable. There exist indefinite metrics which are not locally homogeneous, so that $J(X)$ is nilpotent for all tangent vectors $X$, see, for example, [a5].

If $\{X_1,...,X_r\}$ is an orthonormal basis for an $r$-plane $\pi$, one can define a higher-order Jacobi operator

\begin{equation}J(\pi)=J(X_1)+...+J(X_r).\end{equation}

One says that an algebraic curvature tensor or Riemannian metric is $r$-Osserman if the eigenvalues of $J(\pi)$ are constant on the Grassmannian of non-oriented $r$-planes in the tangent bundle. I. Stavrov [a8] and G. Stanilov and V. Videv [a7] have obtained some results in this setting.

In the Riemannian setting, if $2\leq r\leq m-2$ I. Dotti, M. Druetta and P. Gilkey [a1] have recently classified the $r$-Osserman algebraic curvature tensors and showed that the only $r$-Osserman metrics are the metrics of constant sectional curvature.

References

[a1]	I. Dotti, M. Druetta, P. Gilkey, "Algebraic curvature tensors which are Osserman" Preprint (1999)
[a2]	N. Blažić, N. Bokan, P. Gilkey, "A note on Osserman Lorentzian manifolds" Bull. London Math. Soc. , 29 (1997) pp. 227–230
[a3]	C.S. Chi, "A curvature characterization of certain locally rank one symmetric spaces" J. Diff. Geom. , 28 (1988) pp. 187–202
[a4]	P. Gilkey, "Manifolds whose curvature operator has constant eigenvalues at the basepoint" J. Geom. Anal. , 4 (1994) pp. 155–158
[a5]	E. Garcia-Rio, D.N. Kupeli, M.E. Vázquez-Abal, "On a problem of Osserman in Lorentzian geometry" Diff. Geom. Appl. , 7 (1997) pp. 85–100
[a6]	R. Osserman, "Curvature in the eighties" Amer. Math. Monthly , 97 (1990) pp. 731–756
[a7]	G. Stanilov, V. Videv, "Four dimensional pointwise Osserman manifolds" Abh. Math. Sem. Univ. Hamburg , 68 (1998) pp. 1–6
[a8]	I. Stavrov, "A note on generalized Osserman manifolds" Preprint (1998)