View source for Osserman conjecture

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200901.png" /> be the Riemann [[Curvature tensor|curvature tensor]] of a [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200902.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200903.png" /> be the Jacobi operator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200904.png" /> is a unit tangent vector at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200905.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200906.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200907.png" /> is a self-adjoint endomorphism of the [[Tangent bundle|tangent bundle]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200908.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o1200909.png" /> is flat or is locally a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009010.png" /> symmetric space (cf. also [[Symmetric space|Symmetric space]]), then the set of local isometries acts transitively on the sphere bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009011.png" /> of unit tangent vectors, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009012.png" /> has constant eigenvalues on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009013.png" />. R. Osserman [[#References|[a6]]] wondered if the converse implication was valid; the following conjecture has become known as the Osserman conjecture: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009014.png" /> has constant eigenvalues, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009015.png" /> is flat or is locally a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009016.png" /> symmetric space.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009017.png" /> be the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009019.png" /> is odd, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009020.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009021.png" />, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009022.png" />, then C.S. Chi [[#References|[a3]]] has established this conjecture using a blend of tools from [[Algebraic topology|algebraic topology]] and [[Differential geometry|differential geometry]]. There is a corresponding purely algebraic problem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009023.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009024.png" />-tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009025.png" /> which defines a corresponding curvature operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009027.png" /> satisfies the identities,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009028.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009029.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009032.png" />, there always exists a Riemannian metric whose curvature tensor at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009033.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009034.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009035.png" />; this is a self-adjoint endomorphism of the tangent bundle at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009036.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009037.png" /> is Osserman if the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009038.png" /> are constant on the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009040.png" />. C.S. Chi classified the Osserman algebraic curvature tensors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009041.png" /> odd or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009042.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009043.png" />; he then used the second [[Bianchi identity|Bianchi identity]] to complete the proof. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009044.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009045.png" />, it is known [[#References|[a4]]] that there are Osserman algebraic curvature tensors which are not the curvature tensors of rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009046.png" /> 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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009047.png" /> is nilpotent for all tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009048.png" />, see, for example, [[#References|[a5]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009049.png" /> is an orthonormal basis for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009050.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009051.png" />, one can define a higher-order Jacobi operator

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One says that an algebraic curvature tensor or Riemannian metric is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009055.png" />-Osserman if the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009056.png" /> are constant on the Grassmannian of non-oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009057.png" />-planes in the tangent bundle. I. Stavrov [[#References|[a8]]] and G. Stanilov and V. Videv [[#References|[a7]]] have obtained some results in this setting.

In the Riemannian setting, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009058.png" /> I. Dotti, M. Druetta and P. Gilkey [[#References|[a1]]] have recently classified the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009059.png" />-Osserman algebraic curvature tensors and showed that the only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009060.png" />-Osserman metrics are the metrics of constant sectional curvature.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Dotti,   M. Druetta,   P. Gilkey,   "Algebraic curvature tensors which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120090/o12009061.png" /> Osserman"  ''Preprint''  (1999)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Blažić,   N. Bokan,   P. Gilkey,   "A note on Osserman Lorentzian manifolds"  ''Bull. London Math. Soc.'' , '''29'''  (1997)  pp. 227–230</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.S. Chi,   "A curvature characterization of certain locally rank one symmetric spaces"  ''J. Diff. Geom.'' , '''28'''  (1988)  pp. 187–202</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Gilkey,   "Manifolds whose curvature operator has constant eigenvalues at the basepoint"  ''J. Geom. Anal.'' , '''4'''  (1994)  pp. 155–158</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Osserman,   "Curvature in the eighties"  ''Amer. Math. Monthly'' , '''97'''  (1990)  pp. 731–756</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G. Stanilov,   V. Videv,   "Four dimensional pointwise Osserman manifolds"  ''Abh. Math. Sem. Univ. Hamburg'' , '''68'''  (1998)  pp. 1–6</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  I. Stavrov,   "A note on generalized Osserman manifolds"  ''Preprint''  (1998)</TD></TR></table>