Osserman conjecture

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

Let be the dimension of . If is odd, if modulo , or if , 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 be a -tensor on which defines a corresponding curvature operator . If satisfies the identities,

then 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 of , there always exists a Riemannian metric whose curvature tensor at is . Let ; this is a self-adjoint endomorphism of the tangent bundle at . One says that is Osserman if the eigenvalues of are constant on the unit sphere in . C.S. Chi classified the Osserman algebraic curvature tensors for odd or modulo ; he then used the second Bianchi identity to complete the proof. However, if modulo , it is known [a4] that there are Osserman algebraic curvature tensors which are not the curvature tensors of rank- 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 is nilpotent for all tangent vectors , see, for example, [a5].

If is an orthonormal basis for an -plane , one can define a higher-order Jacobi operator

One says that an algebraic curvature tensor or Riemannian metric is -Osserman if the eigenvalues of are constant on the Grassmannian of non-oriented -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 I. Dotti, M. Druetta and P. Gilkey [a1] have recently classified the -Osserman algebraic curvature tensors and showed that the only -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)