Osserman manifold#Osserman conjecture

Let $M^n$ be a Riemannian manifold. For a point $p\; \backslash in\; M^n$ and a unit vector $X\; \backslash in\; T\_pM^n$, the Jacobi operator $R\_X$ is defined by $R\_X\; =\; R(X,\backslash cdot)X$, where $R$ is the Riemann curvature tensor.{{cite journal |author=Y. Nikolayevsky |year=2003 |title=Two theorems on Osserman manifolds |journal=Differential Geometry and Its Applications |volume=18 |issue=3 |pages=239–253|doi=10.1016/S0926-2245(02)00160-2 |doi-access=free }} A manifold $M^n$ is called pointwise Osserman if, for every $p\; \backslash in\; M^n$, the spectrum of the Jacobi operator does not depend on the choice of the unit vector $X$. The manifold is called globally Osserman if the spectrum depends neither on $X$ nor on $p$. All two-point homogeneous spaces are globally Osserman, including Euclidean spaces $\backslash mathbb\{R\}^n$, real projective spaces $\backslash mathbb\{RP\}^n$, spheres $\backslash mathbb\{S\}^n$, hyperbolic spaces $\backslash mathbb\{H\}^n$, complex projective spaces $\backslash mathbb\{CP\}^n$, complex hyperbolic spaces $\backslash mathbb\{CH\}^n$, quaternionic projective spaces $\backslash mathbb\{HP\}^n$, quaternionic hyperbolic spaces $\backslash mathbb\{HH\}^n$, the Cayley projective plane $\backslash mathbb\{C\}ayP^2$, and the Cayley hyperbolic plane $\backslash mathbb\{C\}ayH^2$.

Clifford structures are fundamental in studying Osserman manifolds. An algebraic curvature tensor $R$ in $\backslash mathbb\{R\}^n$ has a $\backslash text\{Cliff\}(\backslash nu)$-structure if it can be expressed as

:$R(X,Y)Z\; =\; \backslash lambda\_0(\backslash langle\; X,Z\; \backslash rangle\; Y\; -\; \backslash langle\; Y,Z\; \backslash rangle\; X)\; +\; \backslash sum\_\{i=1\}^\{\backslash nu\}\; \backslash frac\{1\}\{3\}(\backslash lambda\_i\; -\; \backslash lambda\_0)(2\backslash langle\; J\_iX,Y\; \backslash rangle\; J\_iZ\; +\; \backslash langle\; J\_iZ,Y\; \backslash rangle\; J\_iX\; -\; \backslash langle\; J\_iZ,X\; \backslash rangle\; J\_iY)$

where $J\_i$ are skew-symmetric orthogonal operators satisfying the Hurwitz relations $J\_iJ\_j\; +\; J\_jJ\_i\; =\; -2\backslash delta\_\{ij\}I$.{{cite journal |author=P. Gilkey, A. Swann, L. Vanhecke |year=1995 |title=Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator |journal=Quarterly Journal of Mathematics |volume=46 |issue=3 |pages=299–320|doi=10.1093/qmath/46.3.299 }} A Riemannian manifold is said to have $\backslash text\{Cliff\}(\backslash nu)$-structure if its curvature tensor also does. 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.

{{unsolved|mathematics|Are all Osserman manifolds either flat or locally rank-one symmetric spaces?}}

The Osserman conjecture asks whether every Osserman manifold is either a flat manifold or locally a rank-one symmetric space.{{cite journal |author=Y. Nikolayevsky |year=2011 |title=Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces |journal=Annali di Matematica Pura ed Applicata}}

Considerable progress has been made on this conjecture, with proofs established for manifolds of dimension $n$ where $n$ is not divisible by 4 or $n\; =\; 4$. For pointwise Osserman manifolds, the conjecture holds in dimensions $n\; \backslash neq\; 2$ not divisible by 4. 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.

• Clifford algebra
• List of unsolved problems in mathematics

{{reflist}}

Category:Riemannian manifolds

Category:Differential geometry