Bundles over projective spaces and algebraic curvature tensors (Q5954167)

Let \(R_{g}\) be the Riemann curvature tensor of the associated Levi-Civita connection of a finite dimensional Riemannian manifold \((M,g).\) A central problem in differential geometry is to describe the relationship between algebraic properties of the Riemann curvature tensor \(R_{g}\) and the underlying geometry of the manifold. A natural algebraic curvature tensor \(R\) is associated with \(R_{g}\). One can study various natural endomorphisms defined by the Riemann curvature tensor and examine the geometrical consequences which follow if these endomorphisms are asssumed to have constant eigenvalues. Such a natural endomorphism was introduced by G. Stanilov and is denoted by \(R(\pi)\). Several authors (\textit{S.~Ivanov} and \textit{I.~Petrova} [Geom. Dedicata 70, 269-282 (1998; Zbl 0903.53016)]; \textit{P.~Gilkey, J.~Leahy} and \textit{H.~Sadofsky}; \textit{P.~Gilkey} and \textit{U.~Semmelmann}; \textit{R.~Ivanova} and \textit{G.~Stanilov} [Symposia Gaussiana, 391-395 (1995; Zbl 0861.53046)]) studied Riemannian metrics where \(R(\pi)\) has pointwise constant eigenvalues; these were classified in dimensions \(\geq 4\) and \(\neq 7.\) In the present paper the author studies the complex analogue of this question by using techniques of algebraic topology. Let us consider a 2n-dimensional Riemannian manifold \((M,g),\) endowed with a Hermitian almost complex structure \(J\). An algebraic curvature tensor \(R\) is said to be complex IP if its associated skew symmetric operator \(R(\pi)\) commutes with \(J\). The eigenvalues of \(R(\pi)\) have the form \(\lambda _{i}\sqrt{-1}\) with multiplicities \(k_{i}.\) If the condition \(k_{0}\geq k_{1}\geq ...\geq k_{\ell }\geq 1\) is assumed, then the following main result is obtained: Let \(R\) be a complex IP algebraic curvature tensor. Suppose that \(\ell \geq 1.\) If \(m\equiv 2\pmod 4,\) then \(\ell =1\) and \(k_{1}=1.\) If \(m\equiv 0\pmod 4,\) then either \(\ell =1\) and \(k_{1}\leq 2\) or \(\ell =2\) and \(k_{1}=k_{2}=1.\) The Stiefel-Whitney classes of the bundles defined by an Osserman algebraic curvature tensor are determined. Finally, the author twisted the tensor of constant sectional curvature by an idempotent isometry to construct complex IP algebraic curvature tensors which exhibit the maximal eigenvalue structure permitted by the just presented main result.