In this paper the notion of
∗
-Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the
∗
-Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of
∗
-Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing
∗
-Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose
∗
-Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.
We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.
Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.
AbstractIn this paper, we study the evolution of the Weyl curvature tensor W of hypersurfaces in n+1 under the mean curvature flow. We find a bound for the Weyl curvature tensor of hypersurfaces during the evolution in terms of time. As a consequence, we suppose that the initial hypersurface is conformally flat, i.e., W =0 at t = 0 and then we find an upper estimate for W during the evolution in terms of time.
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.
On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms