In mathematics, particularly in differential geometry, an Osserman manifold is a Riemannian manifold in which the characteristic polynomial of the Jacobi operator of unit tangent vectors is a constant on the unit tangent bundle.
[1] It is named after American mathematician Robert Osserman.
be a Riemannian manifold.
is the Riemann curvature tensor.
is called pointwise Osserman if, for every
, the spectrum of the Jacobi operator does not depend on the choice of the unit vector
The manifold is called globally Osserman if the spectrum depends neither on
All two-point homogeneous spaces are globally Osserman, including Euclidean spaces
, real projective spaces
, complex projective spaces
, complex hyperbolic spaces
, quaternionic projective spaces
, quaternionic hyperbolic spaces
, the Cayley projective plane
, and the Cayley hyperbolic plane
[2] Clifford structures are fundamental in studying Osserman manifolds.
An algebraic curvature tensor
are skew-symmetric[disambiguation needed] orthogonal operators satisfying the Hurwitz relations
These structures naturally arise from unitary representations of Clifford algebras and provide a way to construct examples of Osserman manifolds.
The study of Osserman manifolds has connections to isospectral geometry, Einstein manifolds, curvature operators in differential geometry, and the classification of symmetric spaces.
[2] The Osserman conjecture asks whether every Osserman manifold is either a flat manifold or locally a rank-one symmetric space.
[4] Considerable progress has been made on this conjecture, with proofs established for manifolds of dimension
For pointwise Osserman manifolds, the conjecture holds in dimensions
The case of manifolds with exactly two eigenvalues of the Jacobi operator has been extensively studied, with the conjecture proven except for specific cases in dimension 16.