It is important to study asymptotic behavior of complete manifold without the assumption of pointwise Ricci curvature bound. A volume growth and curvature decay result was obtained in [4] for various classes of complete, noncompact, Bach-flat metrics in dimension 4. Some similar results were also claimed in [1]. In this note we consider a more general case, that is, the Bach tensor may not nece...

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor h...

We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity condition. Thus we obtain a generalization of the vanishing theorem of Bochner, which has a particularly simple form in dimension 4. As a corollary we obtain that if the canonical Weyl structure on a compact Hermitian surface is non-exact, the symmetri...

Consider {(M, g(t)), 0 ≤ t < T < ∞} as an unnormalized Ricci flow solution: dgij dt = −2Rij for t ∈ [0, T ). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T ) then the solution can be extended over T . Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, the...

A diagram for Bianchi spaces with vanishing vector of structure constants (type A in the Ellis-MacCallum classification) illustrates the relations among their different types under similarity transformations. The Ricci coefficients and the Ricci tensor are related by a Cremona transformation.

Charged particles in the geodesic trajectory of an external gravitational eld do not emit electromagnetic radiation. This is expected from the application of the equivalence principle. We show here that charged particles propagating in an external gravitational eld with non-zero components of the Ricci tensor can emit radiation by the Cerenkov process. The external gravitational eld acts like a...

A local classification of all locally conformal flat Riemannian 4-manifolds whose Ricci tensor satisfies the equation ∇ ( ρ− 1 6 sg ) = 1 18 ds⊙ g as well as a local classification of all locally conformal flat Riemannian 4-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The problem of local descripti...

A cochain complex associated with the vector 1-form determined by the first and second fundamental tensors of a hypersurface M in E"+l is introduced. Its cohomology groups HP(M), called curvature groups, are isomorphic with the cohomology groups of M with coefficients in a subsheaf %R of the sheaf S of closed vector fields on M. If M is a minimal variety, the same conclusion is valid with S^ re...

Let M be a real hypersurface of a complex space form with almost contact metric structure (φ, ξ, η, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator Rξ = R(·, ξ)ξ is ξ-parallel. In particular, we prove that the condition ∇ξRξ = 0 characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic ...